Talk:Logarithm/Archive 1

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This may be outdated, so maybe not worth mentioning in the main article, but I've seen this convention used:

• log == log₁₀
•  ln == log_e
•  lg == log₂

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That convention is still commonly used, especially outside of math. In pure mathematics, ln is virtually the only logarithm used, and (especially in number theory) people write log instead of ln. This has bugged me for a long time, because people even use log in the statement of the prime number theorem without saying which base they're using!

I would hope that in the wikipedia, we always use ln for the natural logarithm; the other two, when they are needed, should be accompanied with a half-sentence explanation.

And yes, I think this is definitely worth mentioning. --AxelBoldt

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Using "log" with no subscript to mean base-10 logarithm, is engineer's notation whose justification is obsolete. Mathematicians nowadays use either "log(x)" or "ln(x)" to mean log_e(x). I prefer the former notation. Michael Hardy 22:16 Jan 26, 2003 (UTC)

But that former notation will confuse all but the most hardcore mathematicians. I'm sure there are more engineers among our potential readers than mathematicians, so it seems prudent to use a notation that's unambiguous for both groups. AxelBoldt 02:52 Jan 28, 2003 (UTC)

I'm a mathematician and I use ln exclusively. I never use log by itself at all, although I sometimes write $\log _{10}$. Loisel 02:54 Jan 28, 2003 (UTC)

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If I am wrong tell me but isn't natural logarithm is an inverse function of expotential function? -- Taku

If you look again, that's the first phrase of the article.

The first phrase says logarithm is an inverse funcion of expotential function. And my question is natural logarithm is an inverse funcion of expotential function. Am I mistaken?

Somewhat confusingly one speaks of the exponential function, meaning e^x, and an exponential function a^x. The inverse funcions are the natural logarithm and the logarithm to the base a, respectively - Patrick 21:24 Jan 28, 2003 (UTC)

Also, the following comment, aside from not being in english, is impertinent:

In computer science, logarithm is used for explaning logarithmic algorithm such as binary search, implicitly assuming the base of logarithm functions is 2.

At the limit, one might add that the base 2 logarithm is sometimes the default one in some computer science texts. Loisel 03:10 Jan 28, 2003 (UTC)

What do you mean by impertinent? Just a few sentences don't hurt the article.

See Michael Hardy's comment below. And, if we pollute the article with snippets of little relevance without any structure, the article becomes impossible to read. Conciseness is a virtue. Loisel 03:33 Jan 28, 2003 (UTC)

I disagree. First of all, the encyclopedia is not a textbook for math classes. The purpose of the article should not only teach about logarithm in pure math term but also mention about usage, even mis-usage. In computer science, the term logarithm pops up and is used with the base 2. I think the article is slightly targeted for math geeks. I think it is more appreciate to mention more about less math-like stuff for general audience -- Taku 03:41 Jan 28, 2003 (UTC)

In any case, the information is already contained in this sentence, taken from the article In most pure mathematical work, log is used to denote loge, in most engineering work, it means log10, while in information theory, it often means log2, which also sometimes is written as lg.

Also, there's no need for name calling. Loisel 03:46 Jan 28, 2003 (UTC)

Why can't we simply use a term computer science or logarithmic algorithm? And also is the sentence like "Whenever a possibility for ambiguity exists, this ambiguity should be resolved by explicitly writing out the base." Point of view? I agree with such a practice but ther is not need to say. -- Taku

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Someone made a nonsensical assertion that nowadays the main use of logarithms is solving equations in which the unknown is in the exponent. I deleted it. Of course, if that were true, then no one would ever think about base-e logarithms, nor differentiate logarithmic functions. This raises a question: If someone reading this page wants to know what logarithms are used for, we could include some material responding to that. That would be a fair amount of work, and I have other things to do, but may a list of links to articles that exhibit examples of the use of logarithms could serve. One such article is Prime number theorem. Surely we could find a variety of such topics. Probably some of them would be on number theory, some on probability, some on analysis of algorithms, some on differential equations of physics, some on information theory, some on entropy in physics, etc. Michael Hardy 03:27 Jan 28, 2003 (UTC)

Also, an extensive "related applications" section can be added to almost all articles in wikipedia, and then you can make all articles three times as big. I don't think that's a good thing. A short, bulleted list might be acceptable, but I would advance with care. Loisel 03:35 Jan 28, 2003 (UTC)

On the other hand, for those with less mathematical background (who might be most likely to consult an article like this one!), we might want to add a paragraph that gives people an idea of why the log idea is useful in general. In a way, "compunded percentage growth" is a kind of log concept; perhaps there is some similar, common analogy that can help explain why log is interesting to mathematicians, engineers, computer scientists, physicists, etc. I'd be more interested in seeing a (short!) paragraph about this kind of thing before how it's used by, say, number theorists. Chas zzz brown 08:15 Jan 28, 2003 (UTC)

I have tried to write the beginnings of such a paragraph. Contrary to Mike above, I do think that logarithms are often used to "solve equations where the unknown appears in the exponent", i.e. to invert exponential functions. Then ln often shows up because its derivative is 1/x, and it is therefore the integrals of 1/x. Maybe we should also mention logarithmic scales widely used in the sciences (pH, dB) and coordinate systems with logarithmic axis scales. Any other things? AxelBoldt 17:31 Jan 28, 2003 (UTC)

I never said logarithms are not used that way; I said it is nonsense to say that that is the main use of logarithms. Michael Hardy 18:32 Jan 28, 2003 (UTC)

link to put in article: Logarithmic measure

You might be interested in seeing the discussion at Talk:natural logarithm. Here is a possible solution that I think is best, although I suspect that most people here will not agree.

First, everyone agrees that "ln" stands for natural log. Even people who don't use the "ln" notation recognize it as such, it poses no misunderstanding, even if very few people (e.g. in mathematics) use it. So, "ln" should remain an accepted usage.

However, the "log" notation appears to have mutually conflicting meanings among different groups of people, and within these groups of people, the usage is fairly standard, common, or at the least, assumed by default. Among mathematicians and many scientists, it's assumed to be natural log, and it is the primary notation used in math for this function (far more than "ln"). Among engineers and others, "log" apparently means log-base-10, and this is assumed to be the default meaning of the notation. And among computer scientists or algorithic analysts, "log" is often taken to mean log-base-2, for a variety of theoretical and practical reasons (log-base-2 naturally (no pun intended!) arises out of the study of algorithms and so forth, in this sense it has a true significance that is not just arbitrary).

Forcing a standard meaning of "log" for all wikipedia users has the effect of alienating the other groups (at least 2 other groups) and it forces them to give up the notation that is standard in their disciplines. Forcing "log" to mean log-base-10 forces mathematicians to give up a notation that is standard and used by 95% of the people. Forcing "log" to mean natural log forces engineers, many scientists, and others to give up being able to use "log" for log-base-10 without specifying the base. (Apparently, advocates of "log" for log-base-10 fail to consider that using "log" in this sense is also using "log" without specifying a base, just like mathematicians use "log" for ln without specifying a base, so the argument that it's bad because "the base isn't specified" is wrong, or at least, this argument can just as well be applied to the common logarithm to argue that people should never use "log" without putting the base 10) Forcing "log" to mean base 2 would be pleasing to computer scientists and so on, but would force mathematicians and scientists to both give up their notation. ANY mandate on the meaning of "log" necessarily forces LARGE numbers of people to give up standard notation. Saying that "log" should be log-base-10 because "more people coming here will be engineers than mathematicians" seems disingenuous and a bit insulting.

So, one possible solution is that the meaning of "log" would not be mandated to any specific meaning, and whenever an article uses "log" without a base, a small line at the top of the article in small font could specify the notation used in that article, with appropriate links to explain the issue. The use of "log" MUST be constant throughout an article, however. And, of course, if someone wants to specify a base explicity, or use "ln", this is perfectly fine. But the "log" notation is used in too many different ways by too many different people.

This solution has the downside that people will not know the meaning of log without looking at the top of the article. BUT THIS PROBLEM ALREADY EXISTS AS THE SITUATION STANDS -- MATH PEOPLE ASSUME IT'S NATURAL LOG AND GET CONFUSED WHEN ENGINEERS AND COMPUTER PEOPLE USE IT DIFFERENTLY, ENGINEERS ASSUME IT'S BASE 10 AND GET CONFUSED WHEN MATH PEOPLE USE IT DIFFERENTLY, AND SO ON AND SO ON. The problem ALREADY EXISTS. Forcing a particular meaning is basically saying, "well, we took a vote, and this many people won, so that's it; everyone else has to adjust". With the solution above, people simply have to be aware of the policy (stated at the top of the article) and then the inconvenience is spread more evenly among the different people who use this notation.

In computing, I've seen lg, or the base is shown explicitly as log₂.

A more flexible solution is the one that people do most commonly in writing; for example "log ... where log is the natural logarithm/where log is the base-10 logarith" Dysprosia 22:56, 8 Oct 2003 (UTC)

Notation

• ln = logarithm to the base e
• lg = logarithm to the base 2
• log = ambiguous (engineers: base 10; mathematicians & scientists: base e; computer scientists: base 2)

ln is universally recognised, so it should stay. Whenever log is used, the base should be specified.

The above seems to be a good summary of what was here before.

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And I agree with the above, but I think that lg should not be used (even if it is the correct notation) because it is too rare. I myself have read some calculus books that say 10 should be the "default" base, and some that say e should be the "default" base.

Brianjd 06:11, 2004 Jun 19 (UTC)

For the old discussion of notation, see Talk:Logarithm/Notation. - dcljr 23:32, 8 Nov 2004 (UTC)

Two things: (1) For me, `lg' is just as ambiguous as `log'. Log base 2 can be referred to unambiguously (I hope) as `ld', but even that isn't very common in areas where base 2 is dominant (computer science, information theory). (2) Specifying an explicit base for `log' is a good idea in many cases, but there are situation when this is not necessary. For example, in computer science O(log n) is unambiguous, since the choice of base does not matter. MarkSweep 08:38, 2004 Jun 20 (UTC)

Most UK A-level textbooks I've seen like to use lg for log₁₀ and ln for log_e (the rest use log for log₁₀, and I've never seen lg used for log₂). Style is up to the article writer, but I propose that, if only one base is used in the article, the first occurrence of log includes the base. Any further occurrences don't need to include the base, since the default base is implied. But, really, we don't mean sin(x°) when we say sin(x), because measuring angles in radians is much more useful (namely, you can use the Taylor series, and sin'(x) = cos(x)). Similarly, log'(x) = 1/x. If we define log = log₁₀, then we're stuck with log'(x) = 1/(x ln 10), which is ugly at best. Of course, log(2) is another special number (namely, it's important when calculating logs of big numbers). --Elektron 15:30, 2004 Jun 30 (UTC)

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Forgive my impoliteness, but let's stop wasting time here. A foolish consistency is the hobgoblin of little minds. Here, people seem to be under the impression that a new symbol needs to be created for each base in common use. Otherwise, how else will the moronic readers tell the bases apart ?

That strikes me as strange. Shannon didn't seem to need "lb" or "lg" or anything like that as a crutch for his readers. His readers just assumed from the context (information theory) that, by "log", he meant the binary logarithm. Somehow, they were able to use context clues to figure out which meaning of the word was in use, just like every other word with multiple definitions.

It also strikes me as strange that, in the rare situation we'd need to be clear about a base, we wouldn't use the clear log_base. Instead, we'd irrationally use abbreviations, which are unclear and used for common, well-repeated situations.

Let's treat log, its abbreviation lg, and its capitalization Log like any other word with multiple meanings and stop arguing over definitions. Thank you. HAND. — 131.230.133.185 08:09, 14 August 2005 (UTC)

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But we should say what notations are, not what they ought to be. As for me, I use ln for natural logarithms, log for indeterminate logarithms, and always specify the base otherwise. (If it is obvious from context, e.g. 2 for Information Theory or musical intervals, I could use simply log, but I'd however specify what the base is once and for all at the beginning of the article.) But the reader should be informed that other people use lg to mean log₂ (that's written even in natural logarithm article, and no doubt it belongs here more than there) and Log for log₁₀.--Army1987 16:10, 14 August 2005 (UTC)

I agree with mentioning in the article that lg is sometimes used for log₂. However, my problem is that after your edit, the article claimed that the notations ln, lg and Log are unambiguous. Well, ln is indeed unambiguous, but lg is ambiguous (can also mean natural or indefinite logarithm) and Log is also ambiguous (can also denote the logarithm as a multi-valued complex function). -- Jitse Niesen (talk) 17:05, 14 August 2005 (UTC)

Yeah. You're right. As for lg I messed up a little, and after my edit the article wasn't even self-consistent. As for Log i wasn't even aware that it had any other use than log₁₀...--Army1987 21:16, 14 August 2005 (UTC)

The article already mentions that "lg" (as well as "log") is used frequently in certain subjects to refer to base two. The simple and concise guiding principle of looking at the context will serve readers new to logarithms at least as well and probably far better than a long listing of hemming and hawing and apologizing over badly applicable tips and tricks. Edits to repeat what was already in the article and other edits ("Log" = log₁₀, etc.) simply serve to crowd the article with the "it's used as ... but not really, because sometimes its used as ...". Concise advice that nearly always works is always better than long-winded, detailed advice that works about half the time.

As you noted, you know exactly what you mean by certain notations ("As for me, I use...") when you write them. You also know exactly how you've experienced them in other works. That is the problem: you (along with me and the rest of us) are nowhere near omniscient. The longwinded version invites people to add their own hasty generalizations to the article, leading to edit wars, and, finally, to the compromise monstrosities that result when trying to include ten people's viewpoints (as already seen in the edit history).

For those that must know how binary logarithms are commonly written, the common shorthand "lg" is already mentioned in the binary logarithm article. In fact, with the links to the detailed articles for each frequently-used logarithm, it should be easy for those who want more information (or want to add detailed information) to go to those articles.

The people doing detailed research on usage will go to the four in-depth articles, anyway. The people new to logarithms, looking to get started using and reading them, don't need to be awash in details from everyone's attempt to show off their knowledge in a prominent place. This is an overview article, after all. — 131.230.133.186 00:24, 16 August 2005 (UTC)

Yes. But you should've fixed the common logarithm page to reflect the change. (Now I did that.) And notice that the note about ln is still there (though I admit ln for log_e is way more common than lg for log₂ or Log for log₁₀. However, at least one sentence for them could be retained...--Army1987 09:47, 16 August 2005 (UTC)

I do not have read the comments in this discussion but what sometimes is true it, that the base does not matter! 212.64.56.124 (talk) 20:13, 26 March 2008 (UTC)

Definition

I thought:

• x^(r/s) = (the rth root of x)^s = the rth of (x^s).

• The logarithmic function is any function which satisfies four conditions:
  1. Domain is all of R.
  2. It is nonconstant.
  3. It is differentiable.
  4. For any real numbers x, y, L(xy) = L(x) + L(y).

• The natural logarithm of x, denoted ln x, is the definite integral of 1/x with the lower limit 1 and the upper limit x.

• The exponential functions are then inverses of the logarithmic functions. This is the only way to define exponentials for irrational exponents.

I have not seen anything like this on Wikipedia. Only that e = lim (n->infinity) (1+1/n)^n and logarithms are inverses of exponents, which is absolute rubbish according to what I have read elsewhere.

Brianjd 08:25, 2004 Jun 18 (UTC)

That anything is the only way to define exponentials of irrationals is surprising, to say the least. Seldom, if ever, is there only one way to do something in mathematics. The power-series

$\exp(z)=\sum _{n=0}^{\infty }{z^{n} \over n!}$

defines exponentials, at least to base-e, for all complex numbers, including irrationals. If you want other bases, you may want to bring in already-defined natural logarithms, and then perhaps you want to proceed as you indicate above, but any assertion that that is the only way is dubious.

Contrary to your assertion, the domain of the natural logarithm function is not all reals, but only positive reals. There is a "multiple-valued" (if that expression can be forgiven) logarithm function defined for all complex numbers except 0. But how did you propose to define the logarithm of 0, if the domain is all reals? Michael Hardy 21:39, 18 Jun 2004 (UTC)

What I have read seems to imply that defining exponentials that way would be seen to be, by most people, as absurd as defining pi by a power series.

Yes, the domain of the natural logarithm is all positive reals. My sources were correct - I posted incorrectly.

Anyway I have not read Wikipedia pages very thoroughly but I remember seeing basically "logarithms are inverses of exponentials; exponentials are inverses of logarithms". Is there any solid definition?

Michael Hardy posted the same response to my talk page. Can we keep the discussion where it belongs?

Also, don't post to someones talk page with the section heading "Definition". When things are taken out of context like that they become gibberish.

Brianjd 05:52, 2004 Jun 19 (UTC)

Exponentials at least to the base e seem to be defined properly...but the proofs seem very complicated.

I have seen a calculus book with much easier proofs of the product rule, quotient rule, and the various proofs related to logarithms/exponentials (logarithms require some knowledge of integration though; as the book used my definition above).

Brianjd 05:55, 2004 Jun 19 (UTC)

I see that my definition is indeed buried in the middle of the natural logarithm page. I think that all common definitions (and mine qualifies as a pretty common definition) should be listed in the intro, or in the first section, with their equivalence proved. Brianjd 06:01, 2004 Jun 19 (UTC)

Don't define the logarithm. As with anything in math, there are too many equivalent definitions to call any of them the definition. Rather, list the properties of the log function, with a side note that they are equivalent. Also, no need to prove this in an article.--Sean Kelly 20:05, 24 Dec 2004 (UTC)

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e is just a number that satisfies a certain property. From first principles (with shamelessly incorrect use of dx),

${\frac {d}{dx}}a^{x}={\frac {a^{x+dx}-a^{x}}{dx}}={\frac {a^{x}a^{dx}-a^{x}}{dx}}={\frac {a^{dx}-1}{dx}}a^{x}$

So we define e as the number such that the first derivative of e^x is e^x. That is,

${\frac {e^{dx}-1}{dx}}e^{x}=e^{x}$

$e^{dx}-1=dx\,\!$

$e^{dx}=1+dx\,\!$

$e=\left(1+dx\right)^{\frac {1}{dx}}$

If we define dx = 1/n,

$e=\left(1+{\frac {1}{n}}\right)^{n}$

Alternatively, if we let f(x) = e^x, it's trivial to prove by induction that

$f^{(n)}\left(x\right)=e^{x}$

Which is where you get the taylor expansion for e^x

The basic definition is, really, just d/dx e^x = e^x. I believe I've just proved that the standard definitions of e are equivalent. --Elektron 16:17, 2004 Jun 30 (UTC)

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Um, could someone put the basic formula of a logarithm in big letters somewhere? Like (but using variables) log(base 2)8 = 3 and 2^3=8 are equal. I know that example is used in the article, but it would be great if a more general one was included that used the big math font. --pie4all88

You mean,

$\log _{b}x=n\qquad {\mbox{means}}\qquad b^{n}=x$

? Also, it shouldn't be called the "basic formula" but rather one way of algebraically defining it.--Sean Kelly 20:05, 24 Dec 2004 (UTC)

Formula moved from article

I moved the formula

$\sum _{j=1}^{k-1}\left({\frac {-2^{-1+j}}{\left(-j+k\right)\,\left(-1+k\right)!}}\right)+{\frac {2^{-1+k}\,\ln(2)}{\left(-1+k\right)!}}=\sum _{n=0}^{\infty }{\frac {{\left(-1\right)}^{n}\,n!}{\left(n+k\right)!}}$

here, because it is not clear to me what it is doing at that place in the article. What connection does it have to the algorithm? Furthermore, I'd like to see a reference (I'm too lazy to check it myself). -- Jitse Niesen 14:55, 17 May 2005 (UTC)

vagueness in need of fixing

This article says:

Some people use the notation: ^blog(x) instead of log_b(x).

Since I've never seen this and I've been around, I'm guessing that the phrase "some people" could be made more precise, e.g., "Researchers in mathematical omphalology use the notation ^blog(x) instead of log_b(x)." Could anyone who knows who uses this notation add that information? Michael Hardy 21:58, 22 July 2005 (UTC)

I have seen this notation in Dutch maths text books for secondary schools, see for instance here (Dutch PDF file, look at pages 6 and 7). However, I'd rather not put that in the article myself since I have really no idea how wide spread it is. -- Jitse Niesen (talk) 23:16, 22 July 2005 (UTC)

Infinity

I vaguely recall from grade school that the log of infinity is infinity -- i.e., it never approaches a specific number. Maybe adding this little tidbit could be helpful, I don't know.

I've added a link to table of limits. This could be added there. --MarkSweep 21:24, 2 August 2005 (UTC)

Napier's bones

Napier's bones are not related to logarithms, whereas slide rules clearly are. I interpreted the relevant sentence to refer to multiplication using logarithms, rather than methods for multiplication in general. Perhaps this should be made clearer. --JahJah 18:58, 19 August 2005 (UTC)

Jitse Niesen, I appreciate the rewrite. However, slide rules use logarithms for multiplication, eliminating the use of tables, while Napier's bones are unrelated, except throught the name. One could equally well mention using the abacus here, as well as many other paper and pencil techniques. --JahJah 19:41, 19 August 2005 (UTC)

True; feel free to improve (I misinterpreted your first comment). What I did not like about the previous formulation ("Napier's bones and slide rules made work even easier: tables weren't necessary.") is that slide rules cannot fully replace tables of logarithms since they are less precise. -- Jitse Niesen (talk) 20:16, 19 August 2005 (UTC)

I have rewritten the passage taking your comments above into account. By the way, I am old enough to have used slide rules, log tables and mechanical calculators when I was in high school. --JahJah 20:43, 19 August 2005 (UTC)

Antilogarithm

Anyone have a reference for the comment that the use of antilog was never widespread? It certainly was common in books of tables when I was in high school in the 1960s.--JahJah 10:37, 20 August 2005 (UTC)

The OED has a citation for antilogarithm (in the sense of this article: it can also mean other things) from 1675. What's the source for the 1800s in the article. --JahJah 10:56, 20 August 2005 (UTC)

The Use of Logarithms is pointless without AntiLogs, because you cannot get back to a proper number without performing the AntiLog calculation from the Log one has calculated from the original numbers.

In Spreadsheets the Logarithm is provided as an '@' function, but never an AntiLog '@' function, consequentally this '@' function is pointless and completely useless.

If the name was changed from AntiLog to Exponentiation, I can find no reference in any make of spreadsheet to an '@' function for exponentiation. Thus rendering spreadsheets still impotent in handling Logarithms properly. I am therefore still left with having to use the 'Four figure tables' I was given when studying for GCE Maths. I find such an omission not only absurd, but downright negligent.

As a notation: who decided that the name should be changed? If it works why fix it? —Preceding unsigned comment added by 84.64.12.230 (talk) 15:41, 24 July 2009 (UTC)

Logs make the calculation of Roots of any size, very easy and could be used in spreadsheets, but for this major omission. —Preceding unsigned comment added by 84.64.98.228 (talk) 15:05, 3 August 2008 (UTC)

I question the notability of colog and antilog. Does anyone know a single modern mathematics book that uses this terminology? The standard notation for the inverse function of $\log _{b}x$ is simply $b^{x}$. --FactSpewer (talk) 06:48, 26 November 2008 (UTC)

Numerous books use antilog (including quite a few current ones); it's still a useful and widely used concept, defined just as you indicate above. Cologs are much less common, but do appear in a few modern books. Also, WP:Notability is not the right concept here; it only applies to article topics, not article contents. Dicklyon (talk) 07:26, 26 November 2008 (UTC)

The concept of antilog is widely used and useful, but it's now called exponentiation! You are right that antilog does occur in books, but the math books in which it appears are mostly from the early 20th century or earlier, with a few exceptions (such as Schaum's outlines, and books discussing antilog circuits instead of antilog functions, and a few others such as a book on acoustics and a book on "Easy math for biologists"). Colog is even more obsolete, as far as I can tell. I looked up colog on Google Books. The first 20 books that came up were from 1918 or earlier, except for a book called "Arithmetic refresher", which is a reprinted book originally published in 1944. Anyway, I just feel that it is wrong for the article to suggest that these are current terminologies; if they are mentioned at all, the article should reflect the current literature by saying that these terms are found mainly in older books, and are now mostly obsolete. --FactSpewer (talk) 00:17, 27 November 2008 (UTC)

It's clear from the book evidence that antilog is still used and is not obsolete. Here are a few hundred from the 21st century. Dicklyon (talk) 07:39, 27 November 2008 (UTC)

Are you arguing the same for colog? Google is powerful enough to find >300 books from the 21st century containing colog, but I wouldn't take this as evidence that it is commonly used. Archaic terminologies do not disappear overnight. If you pick up a random modern textbook, I am relatively sure that you will find neither of these terms. I just think that the Wikipedia page should be honest about this. --FactSpewer (talk) 20:15, 27 November 2008 (UTC)

It's a matter of degree. Antilog is more than twice as common (in books, apparently) as colog, and neither is nearly as common as logarithm. So, no, I don't mean to argue that colog is commonly used, and I should back on the claim on antilog, too. But it is used still, less commonly that logarithm, but not all that rare, and not obsolete. But since you asked if anyone knows a single modern book that uses these terms, I just wanted to point out how easy it is to find one or more. Dicklyon (talk) 05:32, 28 November 2008 (UTC)

OK. I've edited the section of the article in order to reflect this; let me know if you think it's reasonable. --FactSpewer (talk) 19:44, 29 November 2008 (UTC)

recursion algorithm for logarithm

I don't understand how to get the higher n terms in the expression log(x)=n0 +1(n1+1/(n2+1/(n3+... if I use the same method as I did to get n0, all the higher n terms will be equal to 0. (Forgive my ignorance, but I asked people with degrees in mathematics and it's not so transparent to them how to get the higher terms).

Possible answer (by N. Frick):

Well, here is my quick and dirty (non-recursive) algorithm ...

  it is obviously not accurate, but it looks "natural" with nuclear spectrum data 

#define NATLOG_10 2.302585093
float fastPseudoLog( float x )
/* similar to log10 (for x < 40) and similar to logE (for 40<x<250) and similar to log2 (for 250<x<500) and log1.25(500<x<3000)  */    
{ return ( x/(1+x/(2+x/(3+(4*x)/(4+(4*x)/(5+(9*x)/(6+x))))))  / NATLOG_10 );
}

Merging of Natural Logariths with logarithm#unspecified bases

In the natural logarithms article Natural_logarithm it is suggested that this article be combined with the logarithm#unspecified bases article. I am totally against this idea, and would like to know why somebody suggested it. Unless there is a strongly compelling reason to merge, then I suggest that this notice, which links to this page, should be removed. --New Thought 20:48, 25 August 2005 (UTC)

Obviously I referred to the section "Notational conventions", not the whole article, of course. --Army1987 21:21, 25 August 2005 (UTC)

programming languages

I've added a bullet on programming languages to the Logarithm#Unspecified_bases section. It's from memory; someone please check it for accuracy (and whether Pascal should be included). Any corrections should also be applied to the corresponding bullet at Natural_logarithm#Notational_conventions. Thanks, Trovatore 05:13, 9 September 2005 (UTC)

Confirmed for Java [1] Lee S. Svoboda tɑk 18:45, 18 May 2006 (UTC)

rm "clear notation" section

I've removed the following claim:

In order to avoid confusion, a logarithm should sometimes be written to make its argument explicit:

$\!\,\log _{b}(x)+z{\mbox{, rather than }}\log _{b}x+z$

This avoids confusion with:

$\!\,\log _{b}(x+z)$

In fact "log_bx+z" is perfectly clear and unambiguous, and means the same as "log_b(x)+z --Trovatore 16:30, 10 September 2005 (UTC)

In fact I once changed it to:

In order to avoid confusion, a logarithm should sometimes be written to make its argument explicit :

$\!\,\log _{b}(x+z){\mbox{, rather than }}\log _{b}x+z$

This avoids confusion with :

$\!\,\log _{b}(x)+z$

Which is what IMO that was most likely to mean. But MarkSweep reverted my edit.--Army1987 20:50, 11 September 2005 (UTC)

Complex Logarithm

I'm missing remarks on the complex logarithms (different branches etc.). However, I dont feel mature enough in that subject (yet), so I decided not to insert anything regarding this issue. Seraph85 14:26, 20 December 2005 (UTC)

There is now material on this in the complex logarithm article. --FactSpewer (talk) 06:35, 26 November 2008 (UTC)

Useless?

say what? this article is great. I have referred to it at least 30 times this year. I am a composer with occasional need of mathmatics. I cheated my way though high school classes and took none at the university, but am now, out of curiousity learning this stuff. cmon, some of us are interested, if you aren't fine. go read one of the othe 5, 067,798 articles on wikipedia! —Preceding unsigned comment added by 78.102.215.71 (talk) 13:22, 8 October 2008 (UTC)

Amazing how one can write this much info in an online, public and hopefully educational to the masses - in such a fashion that only those who understand the article are those who could have written it to begin with.

I've no doubt this info is terribly mathematically accurate.

It's also terribly useless - I still have no idea why log is important or a reasonable example of it's use that a non-mathemetician could use.

What a waste of WIKI space.

I can buy math books. I don't come here for such info. —The preceding unsigned comment was added by 24.43.125.165 (talk • contribs) .

It depends what your expectations are. Logs are not as important for everyday use as say addition and multiplication. One runs into them if one does numbers a lot, like an engineer, statistician, or somebody working in programming or in financial markets. If you paint landscapes for living, this page will be useless to you no matter what. Otherwise, you are welcome to give constuctive suggestions. Oleg Alexandrov (talk) 17:57, 10 December 2005 (UTC)

I am a student of Electrical Engineering and Computer Science and found this article to be useful. at with Log on my calculator. J C 03:59, 30 January 2006 (UTC)

Your problem is about solving exponential equations. Take the log on both sides, and then this article will become useful, as you will need to use the log properties, like the log of product, the log of power, etc.

I don't think it is fair to blame the article for you not being able to solve the problem. This article is (and should) describe the properties of logs, and is not a textbook chapter on solving exponential equations. Oleg Alexandrov (talk) 21:11, 30 January 2006 (UTC)

I think it might be worthwhile to take a stab at http://simple.wikipedia.org/wiki/Logarithm. Well, I'm not going to, but that would seem to fill the void that some people see. -Rjyanco 00:38, 31 January 2006 (UTC)

Personally I think that the article is exceptionally well crafted. I'm no great mind on math, but the class of the article is obvious.

Still, it does seem that wikipedia should not just be the home of the expert, but also help novices understand concepts. Very basic concepts if need be. 2+2 anyone? That way, more people have the possibility of "graduating" from one wiki to another, progressively becoming more eductated.

Otherwise you stand to lose people, they will be more likely to give up on a subject if it is *always* couched in techanical lanauage. (albeit the authors of this page seem well able to convey their meaning)

All we're asking is some help for the little guy! Come on, surely one of you math minds can create a elegantly simple explanation of logs? (insert challenge here :-) —The preceding unsigned comment was added by 84.203.152.34 (talk • contribs) 20 April 2007.

When was the last time any of the Wikipedia math crowd has ever looked at a regular encyclopedia? If you want to have mathematical definitions for the mathematically fluent, fine, but put them at the bottom. Encyclopedia entries need to accommodate the entry-level crowd particularly, and this page is not at all entry-level-friendly. I have finally come to understand the basic idea of logarithms on my own, and then I came to this page to see what it had to say. The top of the page is terrifying; had I wanted actually to learn something about logarithms, I would have navigated away from it as soon as I could. The intro -- INTRO! -- section contains this: "The base b must be neither 0 nor 1 (nor have an absolute value (modulus) of 1 in the case of the extension to complex numbers, though in this case one only usually considers the complex logarithm), and is typically 10, e, or 2. When x and b are further restricted to positive real numbers, logb(x) is a unique real number." One shouldn't need to be competent in math beyond the logarithm in order to understand Wikipedia's logarithm page.216.185.5.254 (talk) 15:51, 11 February 2008 (UTC)

Yes, the line with the nested parens and long aside about complex numbers was pretty horrible. I added some dictionary definitions and cleaned up that bit. Better? Dicklyon (talk) 04:31, 12 February 2008 (UTC)

Believing that there is some justice in this complaint (which is continued below in "Changes needed"), I expanded and simplified the introductory paragraph. What was there was retained, but spread out somewhat to ease into matters. It is not necessary to enter at the deep end of the pool.

The expansion consists of three parts:

1. The simple statement that logarithms are exponents, with a link to that subject. Hitting someone who has no idea what a logarithm is with anything other than a simple definition all but guarantees that he will give up immediately.

2. The first set of three simple equations that demonstrate that logarithms are non-integer unending decimals.

3. The second set of three equations which lead to log₁₀(64). These quite naturally segué into the statement of the general case, which was already in place.

I believe that the expansion will allow the mathematically weak to (at least) get through the first formula without freaking out.

For those who have worked to bring this article to its present high level of quality, I hope that you will not be immediately turned off by my effort, but will give it a chance, and not instantly opt for a revert. Let's see if it resolves the complaints.

Thank you or your patience.B00P (talk) 06:31, 25 March 2008 (UTC)

I think the lead should be short and concise, not too technical, but not this paedagogical either. Of course, there's nothing wrong with being paedagogical as such, but I think the sort of development now in the lead belongs in a later section, with a title that should not be "Logs for Dummies", but something to that effect. It could follow immediately after the lead, and the lead could explicitly make the reader aware of the existence of this section. But the lead itself should be concise! - Other opinions?--Noe (talk) 08:22, 25 March 2008 (UTC)

pardon me, but isn't there a very simple and most concise definition given in the very first paragraph/section of this article? and then naturally, more detailed information follows, and like any reference material it is the job of the consumer to take or leave it as they need for their purposes. If you are looking for a short, concise definition of log have a look again to the first paragraph. then stop reading, unless you are looking for more complex material. of course if you are looking for a non-mathematical definition of a mathematical concept then this is another issue. The concept of logs isn't impossible to translate into "english" but some concepts in various fields transcend translation (explain the concept of "zebra" to someone who has never seen a horse nor a striped animal). Yet this encyclopedia is referenced by all sorts of people experts and non-experts and should be allowed to function for all, no only for the LCD. In this sense, this article is excellent as the concept is cleanly explained in the exposition and then further developed. After all, you know, someone might be looking for slightly more technical material or references.

I agree. The new lead was too long (see WP:LEAD) and tried to cover too much ground. I have replaced the new lead with a cut-down version of the previous lead. Issues such as fractional exponents, irrational values and an explanation of how to use logarithms do not need to be covered in the lead. Gandalf61 (talk) 13:29, 25 March 2008 (UTC)

A tiny issue in the shortened lead: A major utility of logarithms is that they reduce multiplication to addition - historically, this was a use for logarithms (a motivation, in fact), but today, it's more like an important feature. So perhaps major utility should be changed into something else.--Noe (talk) 08:47, 26 March 2008 (UTC)

Yes, I agree. I have changed "major utility" to "important feature" and added a sentence that links this feature to historical motivation. Gandalf61 (talk) 10:40, 26 March 2008 (UTC)

Work needed?

(continuation of prev discussion; this comment is by Oleg Alexandrov (talk) 04:45, 1 February 2006 (UTC))

That article is yet to be written. I think the article here does need work.--agr 11:59, 31 January 2006 (UTC)

I would like to ask for specific suggestions of what needs work, before work is attempted. Just to make sure there is understanding on what is going on before committing a lot of time and effort. Oleg Alexandrov (talk) 01:38, 1 February 2006 (UTC)

Sorry, I went ahead and did some serious editing. Mostly I reorganized things, with the intro focusing on the basics. I moved the continued fraction algorithm and the proof of the basis change formula to additional topics in logarithms, which I renamed from "using logarithms", a somewhat misleading title. I also tried to make explanations clearer in some places. I think the section on tables needs more work and maybe should be in its own article. --agr 04:06, 1 February 2006 (UTC)

The table reflects the most important properites of the log. While a separate article on that with more details may be needed, I would be against removing the table from the article. Oleg Alexandrov (talk) 04:45, 1 February 2006 (UTC)

I was referring to the subsection "Tables of logarithms", not the table of formulas, which I agree should stay here. I think there is a lot more that should be said about tables of logarithms and that might justify a separate article. Sorry I did not make myself clear. --agr 05:16, 1 February 2006 (UTC)

I see. A common practice on Wikipedia is to create a main article, say "Tables of logarithms", put there a lot of stuff, and then have a shorter version of that at logarithm, while referring to "table of logarithms" for more details. How would that sound? Oleg Alexandrov (talk) 23:52, 1 February 2006 (UTC)

changes needed

The article needs to be divided into two parts - the first part must give an explanation FOR NON MATHEMATICIANS. Mathematicians will not be looking in an encyclopedia to find out about these subjects.Johncmullen1960 05:46, 25 October 2006 (UTC)

I second this wholeheartedly. The concept of logarithms is not so esoteric that the introduction can't be expressed in plain English for the layman. Later sections can dive into as much detail or complexity as desired. --Ahhwhereami 07:37, 28 February 2007 (UTC)

Jainas?

I'm suspicious of the comment about the Jaina mathematicians at the beginning of the history section, since it's uncited anywhere in the article, and since there has been some controversy recently over overzealous crediting of this group. Is there someone who could clear this up? Ryan Reich 15:02, 22 April 2006 (UTC)

New casio

A new casio calculator( The Casio FX83ES - Scientific Calculator) can calculate log's of any base. so there is no need for conversions. page on the webiste.-wolfmankurd

Just a question, but what about those of us who actually could use knowledge like change of base? Or, for instance, on a test where we are not allowed calculators? It's important to know, and I consider it important to keep here. Assuming, of course, that your point by stating this is that you want the Change of Base section removed. T3thys::ben (talk) 02:51, 14 January 2008 (UTC)

Changed the program

This program is easier to understand and faster than the previous one.Wolfmankurd 20:51, 27 April 2006 (UTC)

Sure your program is easier to understand and faster BUT

* How do you calculate 3.5**4.5 without using logarithm?

Rational power can easily be done as roots. In this case, the square root of 3.5**9. Irrational powers as limits or, for approximate answers, linear interpolation. Gah4 (talk) 22:58, 26 April 2010 (UTC)

Also your program does not work! Have you tested it yourself?

$ python prog.pl
Log Base: 2

ERROR!

Ohanian 01:03, 8 May 2006 (UTC)

Definition

The article said that the exponent, or logarithm, indicated how many times to multiply a number by itself. However, an exponent of one does not mean to multiply it by itself one time. That would be an exponent of two. So, I have changed the article to indicate that a positive exponent indicates that a number is multiplied by itself a number of times, with the exponent indicating how many times it's used as a factor. This is more accurate. It leaves the problem of a positive exponent of 1 still being characterized as meaning a multiplication using the number in a multiplication a number of times, in which the number is used only once in the multiplication - an awkward element of this definition, since there is no multiplication at all if the exponent is one. However, it could still be interpreted as meaning it is used as a factor only once (i.e. no multiplication), and I did not judge that complicating the definition with an extensive discussion of this exception was worthwhile - but rather left it to this note to point this out - what is still missing in this improvement in the definition. If someone sees a simple way of describing the exception that fits well in the context, I would welcome its being added. One possibility would be to indicate that the definition is for positive integers greater than one. This might be a good solution. -- Ken Cliffer

I tried to write it as "product of n factors with each equal to b". How's that? :) Oleg Alexandrov (talk) 02:59, 14 May 2006 (UTC)

Oleg, thanks for your input. Please excuse that I'm leaving it to you to make the change, since I think it's better that you agree before I keep changing it (I'm new to this, so please let me know if I should just make the change and comment on it). My comment and suggestion: I like your previous version better than the current one. However, even better, my suggestion would be to change it to this (a little shorter than the previous one, but better still than the current one): "... the product of n factors equal to b ...". I think this is better than "... the product of n copies of b ...," since "copies" seems to imply something more than that it is simply numbers - there's nothing "copied," only factors equal to b. [I had another suggestion here, but on further consideration, I realize I was missing something; therefore I have withdrawn it.]

Okay, I have now read guidelines to be bold (after your invitation to join), so I'll go ahead and make the suggested change.KCliffer 20:59, 14 May 2006 (UTC)

Cool, thanks! Oleg Alexandrov (talk) 00:10, 15 May 2006 (UTC)

I think the current thing about factors is slightly confusing. If you need a definition of b^n, that definitions going to be confusing as all hell. I propose:

"If n is a positive integer, bⁿ means b multiplied by b, n number of times."

I think even this definition doesn't suffice because I don't think a concept of "multiplying sumthing a negative amount of times" really exits - not to metion those needing that definition won't have that concept. Fresheneesz 23:08, 29 May 2006 (UTC)

Your definition suffers from the problem mentioned above. What does it mean to multiply n times? b^2 is b multiplied twice? Does that mean two b's or two multiplications? Oleg Alexandrov (talk) 03:22, 30 May 2006 (UTC)

Hmmm, I guess this is why english isn't used to describe mathematics. Maybe we should forgo the english alltogether, as it seems sufficient explanation has been taken from the article exponentiation. Fresheneesz 17:20, 30 May 2006 (UTC)

Numerical value program

Could somebody write why and how this program works? Not all of us are that familiar with Python. I'd do it myself if I knew Python or was willing to learn it. Lee S. Svoboda tɑk 18:53, 18 May 2006 (UTC)

Sure which do you prefer? Pseudocode or C? Here it is in pseudocode

#
# Function in pseudocode
#
function log(float N,float X)
{  epsilon = 0.000000000001
   integer_value=0
   while (X < 1)
   {
      integer_value = integer_value - 1
      X = X * N
   }
   while (X >= N)
   {
      integer_value = integer_value + 1
      X = X / N
   }
   decfrac = 0.0
   partial = 0.5
   X=X*X
   while (partial > epsilon)
   {
      while (X >= N)
      {
         decfrac = decfrac + partial
         X = X / N
      }
      partial = partial / 2
      X=X*X
   }
   return (integer_value + decfrac)
}

I can take both (Java, my programming language is a C-like languuage). However, I didn't state myself well. Do you know the mathematical formula that makes this program work? Lee S. Svoboda tɑk 16:23, 23 May 2006 (UTC)

Formal definition?

Since it isn't included, I'm going to guess that this is the formal definition:

$\log _{a}x=y\iff a^{y}=x$

Or, verbosely: log base a of x equals y if and only if a to the yth power equals x.

Can anyone verify this? If so, anyone can go ahead and add it to the article upon verification. -Matt 05:44, 20 May 2006 (UTC)

This is a good way to understand the logarithm, but it's not usually used as the formal definition from first principles. The standard excuse for why it's not used is that defining exponentiation is slightly tricky for irrational exponents, though it's my opinion that this excuse has no teeth. The usual formal definition is as an integral. -lethe ^{talk +} 23:14, 29 May 2006 (UTC)

Recent trimming

I removed some stuff from this article, because I think the Python code is unhelpful to anybody not knowing python (a simple explanation on how the algorithm works and when it works would do a much better job). I also removed the recently inserted series expressions as I feel they were not so relevant, and the article was getting too big anyway. Wonder if there are nay comments on that. Oleg Alexandrov (talk) 17:42, 23 May 2006 (UTC)

I would have preferred the python program replaced by a mathematical formula, but do not disagree.

Lee S. Svoboda tɑk 20:25, 25 May 2006 (UTC)

Manual Calculation

Does any one have any information on how the vast tables of logarithms were calculated. Clearly, it is not a simple task or tables would not have been necessary. 68.6.85.167 21:52, 29 May 2006 (UTC)

It's nice to see someone has the same doubt. I'd suggest to add at least a line on how those logarithm tables were calculated, they certainly weren't made using a calculator, and else than using one of those, I have no idea on how to manually calculate a logarithm, neither I have heard an explanation of it, ever. Pentalis 00:24, 14 July 2006 (UTC)

Here is a link to a website that hosts an article explaining how to calculate logarithmic table unfortunately I think that the article its self is copyrighted and to even read the article you need to log in on the website. http://www.jstor.org/view/00255572/ap060497/06a00150/0 Motobiker91 02:16, 15 November 2007 (UTC)

In the "Feynman Lectures on Physics", I believe volume 1, he explains it through successive square roots of 10. First calculate the square root of 10, or 10**0.5. Next the square root of that, 10**0.25. Continue on depending on how many decimal places you need. There is a pencil and paper algorithm for square root. (This is also why base 10 logs became popular.) Continuing, once you calculate sqrt(10) you have one entry in a log table, as log10(sqrt(10))=0.5. If you multiply 10**0.5 and 10**0.25 you get 10**0.75. So after a small number of square roots all you need is multiplication. Gah4 (talk) 09:57, 26 April 2009 (UTC)

Base of log, ld, lg, ln

The anon editor 87.193.43.142 indicated that ld should be base 2 and lg base 10. He may have a point. The d then stands for dual. I have never seen lg used for base 2.

The section about the various conventions is quite confusing. It might be condensed along the following lines. The abbreviation ln is universally used to stand only for base e and log10 for base 10. The name log can be base e (maths) or base 10 (engineering, including calculators and computer languages). The name ld is sometimes used in computer science for base 2. The name lg is sometimes used for base 10. −Woodstone 19:39, 11 August 2006 (UTC)

The name log can be base e (maths) or base 10 (engineering, including calculators and computer languages). This is wrong. I've never used a programming language where log means anything other than the natural logarithm. Fredrik Johansson 20:22, 11 August 2006 (UTC)

Where do you find that in the article? All I see is the part that says that in most commonly used programming languages, "LOG" means natural logarithm, and the part that says that on calculators "LOG" means base-10 logarithm. Michael Hardy 23:16, 11 August 2006 (UTC)

I was quoting Woodstone. Fredrik Johansson 23:54, 11 August 2006 (UTC)

Dubious notation

The article has this:

• Also frequently used is the notation ^blog(x) instead of log_b(x).

I have never seen that notation, and I've been around. Who uses it? If it prevails in some particular field, the article should say which field. Michael Hardy 23:17, 11 August 2006 (UTC)

I haven't seen it either (though admittedly I haven't been around :-). By the way, the symbol log_b x can also mean a b-foldly nested logarithm (MathWorld). This should perhaps be covered in the article. Fredrik Johansson 02:13, 12 August 2006 (UTC)

Whoops! I just happened to look in one of my old calculus textbooks and it does use the notation ^blog(x). How strange that I didn't remember that. This presumably means that the notation is common in Europe. Fredrik Johansson 08:54, 20 August 2006 (UTC)

I had a similar but opposite experience. Before looking at this wikipedia article, I could not remember ever having seen the log_b notation. But a quick look in Google confirms it's the dominant notation. In my high school and university training we always used the ^blog notation. One slight advantage of that notation is the elegance of the formula for change of base: ${}^{a}\log b={}^{a}\log c\cdot {}^{c}\log b$. −Woodstone 09:19, 20 August 2006 (UTC)

Frederik, which calculus book are you referring to? Woodstone, where the high school and the university you refer to? Michael Hardy 18:30, 22 August 2006 (UTC)

If it helps, Persson & Böiers, Analys i en variabel. (In Swedish.) Fredrik Johansson 20:27, 22 August 2006 (UTC)

---

can someone explain where this equation is from....Log a^b in base a^c = b/c...in example log 6^5 to base 36....would mean log 6^5 to base 6^2 which therefore with the equation would result in the solution being 5/2...which is 2.5. -By Babak S. yaaa diiiig

(a^c)^(b/c) = a^(c*(b/c)) = a^b QED. --agr 04:28, 22 August 2006 (UTC)

History is unreferenced

Th whole of the history section is unreferenced. It's like 'take-my-word-for-it'. It's not encyclopaedic, please reference.

History section

I've removed an entire paragraph in the history section regarding the contributions of Jaina and Muslim mathematicians. There does not seem to be a source for this at the moment. In some sense, I think what was written falls in the category where we should have zero information for the moment. See ([2][3][4]) for some background. In the meantime, I'm going to see if I can get in touch with a historian and perhaps find some resources to support the claim - I think trigonometric tables were tabulated quite early in the history of mathematics, but I am not so sure about logarithms. --HappyCamper 01:01, 7 September 2006 (UTC)

.

A great way to explain something so that most people will not understand it. —Preceding unsigned comment added by Wndwshppr (talk • contribs)

Could you be specific? Michael Hardy 20:14, 31 October 2006 (UTC)

---

Johannes Kepler

When talking about the history of logarithms, you cannot exclude Johannes Kepler, Because he was the first to derive logarithms purely mathematically. This unsigned comment was added by 71.195.204.65 23:51, 14 November 2006, and later modified by somebody else.

Quite right. I added a line about it (and then noticed this comment). B00P (talk) 07:42, 24 March 2008 (UTC)

Computing section

When discussing how to compute logarithms it might be a good idea to explain how ocmputers compute log(x) - logarithm with base e.

The basic outline can be as follows:

First, the number x is typically a floating point number of the form m * 2^n where m is mantissa and n is an integer exponent.

log(x) = log(m) + n * log(2)

log(2) is a constant - can be computed once and for all.

What is left is to compute log(m) of a relateively small number m.

Should probably also here add in how the series for logarithm is developed.

log(1 + z) can be written as the integral of 1/(1 + z) and is the integral of the series 1 - z + z^2 ... (include a link to power series here) and so log(1 + z) = z - 1/2 z^2 + 1/3 z^3 etc...

Similarly, log(1 - z) = -z - 1/2 z^2 - 1/3 z^3 etc...

If you were to add those two you would simply get log(1+z)(1-z) = log(1-z^2) = -z^2 - 1/2 z^4 etc which isn't very interesting since you get the same if you replace z with z^2 in log(1-z). However, if you instead subtract them you get:

log(1-z)/(1+z) = 2( z + 1/3 z^3 + 1/5 z^5 ...)

Thus, if you can find an x such that (1-z)/(1+z) = m you will be able to compute log(m) by that series.

I.e. 1 - z = m + mz or z = (1-m)/(1+m). If m is very small as for denormal numbers the divisor here is close to 1. For normal floating point numbers you can adjust m to be in the range 1 <= m < 2 or 1/2 <= m < 1 and in that case you will get a fairly low value for z which allows you to compute log(m) fairly rapidly using only a few terms in the computation unless you want very high precision.

Once log(m) is computed - and log(2) can also be computed in this manner - you can then compute log(x) = log(m) + n * log(2) by a simple multiply and add.

log10(x) can then be computed using the formula, log10(x) = y means that 10^y = x and taking log() on both sides gives: log(x) = y * log(10) so log10(x) = y = log(x) / log(10) or in general log(x,b) = log(x) / log(b). where log10(x) = log(x,10) and log(x) = log(x,e). log(b,b) = 1 for any valid base b.

salte 12:53, 1 December 2006 (UTC)

Building on what I wrote on Friday I suggest we include something like this in the main page.

Computation of logarithm in computers

As all other logarithms can be computed based on any other, it is natural to first define the logarithm for a specific base. Using e as this base one can then easily compute other logarithms based on that function.

log(x) can be computed using a series. This series is found by first looking at

${\frac {1}{1-z}}=1+z+z^{2}+z^{3}+z^{4}...$

Integrating on both sides gives:

$-\log(1-z)=z+{\frac {1}{2}}z^{2}+{\frac {1}{3}}z^{3}+{\frac {1}{4}}z^{4}+{\frac {1}{5}}z^{5}...$

or

$\log(1-z)=-z-{\frac {1}{2}}z^{2}-{\frac {1}{3}}z^{3}-{\frac {1}{4}}z^{4}-{\frac {1}{5}}z^{5}...$

Substituting -z for z gives us

$\log(1+z)=z-{\frac {1}{2}}z^{2}+{\frac {1}{3}}z^{3}-{\frac {1}{4}}z^{4}+{\frac {1}{5}}z^{5}...$

Adding the first and the third together and using

$\log A-\log B=\log {\frac {A}{B}}$

gives us:

$\log {\frac {1+z}{1-z}}=2(z+{\frac {1}{3}}z^{3}+{\frac {1}{5}}z^{5}+{\frac {1}{7}}z^{7}...)$

This is the basic series. However, it is not very useful for large values of z and even if you can compute log(x) for any x such that abs(z) < 1 it might still be useful to next consider how floating point numbers are represented in a computer.

Typically, floating point numbers are represented using three numbers: sign, mantissa and exponent. The sign is either +1 or -1 and is typically represented with one bit using 0 for +1 and 1 for -1.

mantissa is a number that represent the digits of the number. The exponent represent where the binary point (similar to the decimal point for decimal numbers) is placed.

Thus, the value of a floating point number is then:

$value=sign\times mantissa\times base^{e}xponent$

The base is always a fixed constant and for most modern computers it is 2. Thus, to compute log(x) when x is of this form it is:

log(x) = log(sign) + log(mantissa) + exponent * log(base)

Here, the sign must be positive, the log function for real numbers is not defined for negative x, thus the sign is always +1 and log(1) = 0 so this contributes nothing.

What is left is the mantissa and the exponent. The exponent is known by simply extracting it from the number. So what we need to compute is log(mantissa). For most numbers the mantissa is always in the range 1 <= mantissa < 2 (or 1/2 <= mantissa < 1). This means that our z will actually always be 0 <= z < 1/3 (or -1/3 <= z < 0) and so the earlier mentioned series can be used to compute log(mantissa). log(x) is then simply log(mantissa) + exponent * log(base) and using 2 for base this is then easily computed.

log(2) can also computed using the above series as you get z = 1/3 or

$\log 2=\log {\frac {1+{\frac {1}{3}}}{1-{\frac {1}{3}}}}=2({\frac {1}{3}}+{\frac {1}{3}}({\frac {1}{3}})^{3}+{\frac {1}{5}}({\frac {1}{3}})^{5}+{\frac {1}{7}}({\frac {1}{3}})^{7}...)$

Thus, if x is a positive floating point number with exponent n and mantissa m, then we can compute log(x) as follows:

$\log(x)=\log(m2^{n})=\log m+n\log 2$

From m, we compute z, as

${\frac {1+z}{1-z}}=m<=>1+z=m-mz<=>m-1=z(m+1)<=>z={\frac {m-1}{m+1}}$

From this we compute log(m) using z in the series above and then compute log(x).

For complex arguments $z=x+yi$ it is probably best to convert x and y to polar co-ordinates and then use the identity:

$z=x+yi=re^{{\theta }i+k2\pi }$

so $log(z)=log(r)+{\theta }i+k2\pi$

For complex arguments, the log function becomes multivalued in that any integer value for k gives a valid value since $e^{2{\pi }i}=1$.

The values $r$ and $\theta$ can be found as: $r=|z|={\sqrt {x^{2}+y^{2}}}$ and $\theta =\arctan 2(y,x)$ where y is the imaginary part and x is the real part of the complex number z.

---

I will move this to the main page 3 days from now if that is ok with everyone.

salte 10:11, 4 December 2006 (UTC)

Complex numbers and other domains

The section above indicates that complex logs will be added to the article, which is good. As it stands it's not clear what the domain of the logs, is though positive Real is assumed. Other domains, such as Complex are possible. I think there may be other possibilities too, such as for modular arithmetic, and groups e.g of matrices. David Martland 12:56, 7 December 2006 (UTC)

Added more on how to compute logarithm in computers.

Ok, I did what I threatened to do earlier and added in some stuff on computing logarithm in computers. Also added some information on how to obtain the series for logarithm. It is sketchy though but I think that is ok as this page is about logarithm and not on inverse function or integration of a series.

salte 12:53, 8 December 2006 (UTC)

Logarithms are not the only isomorphisms between R and R+

I've just removed, twice, the following (highlighted) statement:

For each positive b not equal to 1, the function log_b (x) is an isomorphism from the group of positive real numbers under multiplication to the group of (all) real numbers under addition. They are the only such isomorphisms.

That's incorrect (and an exam question I've seen at least twice). The trick is that instead of considering the groups just as abelian groups, we can equivalently (proof left as exercise) consider them as Q-vector spaces, and as a Q-vector space, the reals have many automorphisms — in fact, more than there are real numbers, so if there is one isomorphism between R and another group, there are more than there are real numbers, so they can't all be of the form log_b.

RandomP 21:33, 23 December 2006 (UTC)

Well, I put it back again before reading your explanation, and I must confess I have no idea what you're talking about. Can you provide either an example or a condition that would make the statement true, so that we can say something useful in place of what you're telling us is wrong? Dicklyon 21:51, 23 December 2006 (UTC)

What it's talking about is that in standard mathematics, you can define a bijective function f from the positive reals to the real numbers such that f(a*b) = f(a) + f(b) for all a, b positive real, but which isn't otherwise like a logarithm function. Unfortunately (from the pedagogic point of view) it isn't possible to really explicitly construct such a function, as far as I know; read axiom of choice for more information.

The condition you're looking for might be continuity or monotonicity; both make the statement correct. I'm not sure either statement is particularly useful.

RandomP 22:31, 23 December 2006 (UTC)

Jost Bürgi

I believe it violates NPOV to state that Jost Bürgi invented logarithms "first". Since he did not publish until four years *after* Napier, by which time Napier's work was widely known, de facto Bürgi does not deserve credit for being "first". In addition, it is not clear that Bürgi was first in any sense, since Napier worked on his version for 20 years before publishing. Is there documentary evidence that Bürgi started before 1594?

The neutral point of view is that Bürgi *independently* developed logarithms.

I have edited both the logarithm and Jost Bürgi Wiki articles to reflect this, giving Bürgi credit for independently inventing logarthms, but not for being first.

Nwbeeson 18:28, 4 January 2007 (UTC)

Function vs Number

I think that the logarithm can either be considered a function that yields a number or the number itself, depending on the convention. Perhaps it is best to include that in the lead paragraph? (See [5] and [6] where it is defined as a function; some other sites use "logarithmic function" instead of logarithm function though). —Mets501 (talk) 19:31, 26 January 2007 (UTC)

For people who understand numbers and functions, it's fine as it is; the lead is clearly desribing a function. However, for people who don't see that, perhaps you're right that we need to work the word function into it explicitly. But not carelessly, like your second ref that says "Logarithms are one of the functions...". And we might need to decide whether to describe one function of x and b, or an infinite number of functions of x, parameterized by b. We should perhaps look for a better way to put it. Dicklyon 22:15, 26 January 2007 (UTC)

Ah, I just noticed the point of your comment being Michael Hardy's removal of your contribution, which was this lead:

In mathematics, the logarithm is the function that is defined as the inverse of the exponential function. The logarithm of a number x in base b is the number n such that x = bⁿ, ...

which he replaced with edit comment: (The logartithm of a specified number to a specified base is NOT a function; it is a number. A logarithmic function is a function. I'll be back later to add more.) with this older version:

In mathematics, the logarithm of a number x in base b is the number n such that x = bⁿ,

I agree with your point that the log is both, and there's no conflict in your longer lead. Michael's comment is true, but I don't see how it applies, since you did not write that a the log "of a specified number to a specified base" is a function, but that "the log is the function that is defined s the inverse...". Sounds right to me. The second sentence refers to "logarithm of a number x in base b" which means, to anyone who knows about functions and numbers, the number the results from applying the function. Just like you could define square root as the inverse function of squaring, and then the square root of x is a number.

Is there a clearer way to put the fact that the term logarithm can refer both to the function generally and to the number that you get get by applying the function? Michael, ideas? Dicklyon 22:29, 26 January 2007 (UTC)

I think the explanation that I put at the beginning should come first, and the "function" idea later. Michael Hardy 00:50, 28 January 2007 (UTC)

Basically exactly what I was going to write before I got an edit conflict :-). Let's find out what Michael thinks... —Mets501 (talk) 22:33, 26 January 2007 (UTC)

Right after the very short introductory paragraph, I've added a new section titled Logarithmic functions to treat these issues. Michael Hardy 23:07, 26 January 2007 (UTC)

Micheal, from your comment and your edits, it seems you are disagreeing with us, and denying that the logarithm is a function or that "the logarithm function" is what it is called. And I disagree with your interpretation of "logarithmic function". A logarithmic function is any function that has the shape of a logarithm function. For example, in a MOSFET in weak inversion, the voltage from gate to source is a logarithmic function of the source-drain current. But it can NOT be written as just a log base b of I, because it needs a dimensional scale factor. The function is logarithmic, but it is not the logarithm function as defined in the article or as conventionally defined elsewhere. So, I think you've misinterpreted the words "logarithmic function" that you've read some place. Do you have a source for your interpretation? Dicklyon 23:24, 26 January 2007 (UTC)

What I wrote is entirely standard usage among mathematicians and---I think---others. I don't see that you shown that it conflicts with the example you've mentioned, involving dimensional quantities. And even so, the definition does not imply that there are no variations on the theme of the definition---of course there are; that always happens. "That I've read some place"?? I don't know how many books I've read about logarithmic functions in. Hundreds or thousands, probably. However, I will invite other Wikipedia mathematicians to look at this article with your points in mind, by mentioning it at Wikipedia talk:WikiProject Mathematics. Michael Hardy 23:52, 26 January 2007 (UTC)

Thanks, that kind of review will probably be useful. I'm not a mathematician per se, but to me "logarithmic" is a generic descriptor, while logarithm is the name of a function. I checked google book search, and found "logarithmic function" a bit more common than "logarithm function", but of course the latter doesn't begin to capture all the ways that people refer to the logarithm as a function. And of course if you've seen it in hundreds of books, finding one to share shouldn't be a burden. Dicklyon 00:28, 27 January 2007 (UTC)

Here, I'll respond for Michael. He's had a trying day. I've got a few books to share, Dick. My copy of Webster's Third New International Dictionary says that a logarithm is "the exponent that indicates the power to which a number must be raised to produce a given number <if B² = N, then 2 is the logarithm of N (to the base B)>". My old (1976) copy of Encyclopædia Brittanica says "logarithm, the exponent or power to which a base must be raised to yield a given number. In the expression b^x = N, for example, if b is the base and equal to 10 and N a number, equal to 100, then x is equal to 2 and is said to be the logarithm of 100 to the base 10." So it appears that the common usage of the word logarithm is that a logarithm is a number (an exponent, or a power), just as Michael maintains.

My even older (17th edition, 1969) copy of CRC Standard Mathematical Tables contains a table headed "Seven place mantissas of common logarithms", and another entitled "Table of natural or Naperian logarithms". Of course, if the word logarithm meant "a function", these tables would be headed "Seven place mantissas of the common logarithm" and "Table of the natural or Naperian logarithm". But they are not so labeled, because a logarithm is not a function; it is a number. The tables contain logarithms (plural), and those logarithms are numbers. If logarithms are numbers, then a logarithm is a number.

The source of your confusion is obvious. Since mathematicians commonly write f(x) =ln(x), it is easy to conflate the notion of a function (f) with the abbreviation (ln) and then to conclude that ln = logarithm to the base e. In fact, in standard mathematical notation, the abbreviation (ln) stands for "that function which assigns to every positive real number x its logarithm to the base e". Since people don't like such lengthy locutions, they often resort to shorter informal modes of expression. One ought not misinterpret the meaning of an abbreviation.

In fine, I will defend the truth of the statement "a logarithm is a number" as a matter both of grammatical precision and mathematical exactitude. DavidCBryant 02:59, 27 January 2007 (UTC)

It is completely standard to use the phrase "the logarithm" to refer to the function. Two examples from Google books:

1. "The logarithm is the function log: (0,oo)->R such that for any x>0, exp(log x) = x" Mathematical Analysis and Applications: An Introduction By J V Deshpande

2. "The natural logarithm is defined, as in the real domain, as the inverse of the exponential function" Elements of the Theory of Functions By Konrad Knopp

Moreover, the phrase "branch of the logarithm" only makes sense if logarithm is considered a function, not a particular value.

In short, I think that "logarithm" is commonly used to refer to a function, and also commonly used to refer to a value of that function. You might find that unfortunate, but the article might as well just point out that both meanings are used in practice and move on. CMummert · talk 03:47, 27 January 2007 (UTC)

David and CM, it seems obvious to me that you're both right. Thanks for the citations. I'd just like to point out that no reliable source showing one usage contradicts the other usage, unless they do so explicitly. I haven't seen any such explicit contradiction, and so it makes sense for wikipedia to represent BOTH common usages. Right? Dicklyon 06:43, 27 January 2007 (UTC)

The common practice on WP is to just describe how the terms are commonly used in the real world. For example, see the first para of uncountable set. This is a better service for readers than choosing the "correct" terminology and shunning all the others, because readers will come here from many diferent backgrounds, and will likely be confused if the terminology here seems to contradict the terminology they have already seen. It would be nice if real-world terminology was standandardized, grammatically correct, and logical, but it often fails these criteria. CMummert · talk 13:45, 27 January 2007 (UTC)

I don't see exactly what this discussion is about. It relates to terminology somehow, but I don't see quite how. I would use the words the natural logarithm to refer to the inverse function e^x, which is a synonym for the natural logarithm function. But it hurts my ear to read logarithm function in general, with the word logarithm used as an adjective, so I would write the synonymous phrase logarithmic function for euphony. Maybe somebody can state exactly what the disagreement is about? CMummert · talk 01:40, 27 January 2007 (UTC)

Sure: basically, the disagreement is whether the word "logarithm" by itself refers to a function, value, or both. —Mets501 (talk) 01:44, 27 January 2007 (UTC)

It can be used to refer to either. The natural logarithm is the inverse of the exponential function [7]. The natural logarithm of e is 1. Like many classical subjects (including trigonometric functions as well), the terminology is nuanced. CMummert · talk 01:56, 27 January 2007 (UTC)

Exactly, that's what I was saying in response to Michael Hardy's edit summary (The logartithm of a specified number to a specified base is NOT a function; it is a number). —Mets501 (talk) 01:59, 27 January 2007 (UTC)

Let's wait to give everyone else a chance to respond before pressing the issue further. The main question is the nature of any disagreements about terminology. CMummert · talk 02:08, 27 January 2007 (UTC)

Here's me jumping in. I will begin with a little history, a little etymology, from a fun source. At http://members.aol.com/jeff570/l.html, we find that Napier first used "logarithmus" around 1614 for a number, and its English equivalent, "logarithm", occurs soon after. Contrast that with "logarithmic function", which does not appear for another 200 years.

Let's move on to a mathematical distinction, which I will illustrate with "polynomial". The expression "4x³−3x" is a polynomial, formally a member of the ring Z[x]. It is an algebraic object, like a complex number, which we can play with in various ways. One of those ways is evaluation, substituting a number for x to get a number for the expression. Using this we can define the "polynomial function" p:R→R, where p(t) equals the evaluation of the aforementioned polynomial at t. I claim that we need to be equally careful about distinguishing a "logarithm" from a "logarithmic function", at least sometimes.

Which brings us to Wikipedia. Mathematicians abuse notation, and some writers are painfully sloppy. Non-mathematicians are rarely sensitive to our distinctions, and use words to suit themselves. Our job is more to describe than to dictate, but still we wish to set a good and helpful example of careful mathematical usage. We have adequate storage and bandwidth for two articles. So I would urge that "logarithm" be defined as a number, with a note that it may sometimes be shorthand for a "logarithmic function", especially the inverse of e^x. --KSmrq^T 17:27, 27 January 2007 (UTC)

Just to make things more complicated, in the (rather common) phrase "when logarithms were discovered", what was discovered wasn't really a function (and most definitely not a family of functions - other bases came later!), but the idea that the rather primitive integer-valued logarithm function defined on powers of 10 could be extended to a real-valued logarithm function defined for all positive real numbers.

I suspect that the hypothetical reader arriving at this article after having read that phrase would be rather disappointed, particularly if they are young enough not to have practiced long division extensively, and thus less likely to appreciate the significance of replacing it by table lookups and subtraction.

RandomP 02:19, 27 January 2007 (UTC)

It has been over a year since Jan. 20 2006 when this diff got rid of the log being a function and made it a number. For most of 2006, however, it was an "operation"; is that even sensible? Dicklyon 02:56, 27 January 2007 (UTC)

Context!!

OK, I thought this was obvious, but several people are not getting it:

This is context-dependent. In some contexts, it is incorrect to say the logarithm is a function; in other contexts, it is correct. I was pointing out that in one particular context, it is incorrect. I wrote:

"The logartithm of a specified number to a specified base is NOT a function; it is a number. A logarithmic function is a function."

So a bunch of people come along and point out, correctly, that in some contexts it is correct to regard the logarithm as a function. So far, so good. But then they go on to say that that proves I'm wrong. Michael Hardy 00:45, 28 January 2007 (UTC)

As you know, you're not wrong there. I don't see any actual disagreement in the discussion here about what is going on. It would be better to for those interested to just work on the article and let anyone who has an issue with what it says raise that issue here. CMummert · talk 01:21, 28 January 2007 (UTC)

Michael, if you'll review the comments, you'll see that I agreed you were right in that comment. I just didn't see how the comment supported the edit. I don't find the word "wrong" in the discussion, so you leave me wondering what you're referring to. Was it that I had a broader interpretation of the phrase "logarithmic function" than you had, as including more than the logarithm function? Dicklyon 01:59, 28 January 2007 (UTC)

The logarithm function

Looks like Riemann thought "logarithm" was the name of a function: Elements of the Theory of Functions of a Complex Variable: With Especial Reference to the Methods... by Heinrich Durège, 1896, says "We designate, after Riemann, by the name logarithm a function f(z), which has the property that..." google book. Dicklyon 07:02, 27 January 2007 (UTC)

You're missing the point! See my "context!!" comments above. Michael Hardy 00:47, 28 January 2007 (UTC)

Well, Riemann wrote most of his papers in German, and some in French, so what he probably thought of was eine logarithmus (German was his native tongue). Besides, the title of the chapter to which you linked is "Logarithmic and Exponential Functions", leading me to think that Durège (or maybe his editor) was prim and proper part of the time, and not so careful at other times.

Maintaining consistency in Wikipedia is tilting at windmills. I'm sort of a stickler for grammar, though, and I try to think clearly. A logarithm is a number. A logarithmic function assigns a logarithm to each positive real number. The phrase "logarithm function" is an abuse of the attributive noun. That being said, I really don't care how this article misuses the English language. I do see some glaring inconsistencies, though, which probably ought to be rectified.

• The introduction suggests that logarithms can be calculated over a complex base. In practice this is almost never done; mathematicians invariably use the base e in complex analysis. The section "The logarithm as a function" states that the base b "must be positive and must differ from 1", and this is inconsistent with the "roots of unity" language in the introduction.
• The caption under the illustration is worded very poorly. In particular, the locution "Logarithms of all bases pass through the point (1, 0) ..." suggests that logarithms are functions, and not numbers; this is inconsistent with the introductory paragraph.
• Perhaps a reasonable compromise is to use phraseology such as "Strictly speaking, a logarithm is a number ... Informally, the word 'logarithm' is often used as shorthand for 'logarithmic function' ..." DavidCBryant 14:38, 27 January 2007 (UTC)

We say "the logarithm of five is 0.69897" (or 1.6094 or whatever, depending on the base). With respect to 5, "logarithm" is a function. With respect to 0.69897, "logarithm" is the value of that function, a number. It's no different from saying the "cosine is 0.3765" in one case, and "0.5 < cos x < 0.75" or "the cosine of x is between ½ and ¾" in another. Gene Nygaard 15:11, 27 January 2007 (UTC)

Exactly. Anything we say that would demote the function to be a secondary or informal meaning is unacceptable. The log is a function. The log of a number is a number. Just like cosine, square root, exponential, etc. Dicklyon 17:01, 27 January 2007 (UTC)

OK, fine. The physicists are going to dictate terminology to the mathematicians. It's OK by me. There are very deep mathematical reasons for distinguishing between numbers and relations (including functions). But, as I've already stated, maintaining consistency in Wikipedia is tilting at windmills. Don Quijote I am not.

This discussion is starting to remind me of QM 102, with Rochus von Vogt. Herr von Vogt would scribble some big quadruple integral over all space and time up on the blackboard, and ask if there were any questions. I'd say something like "Ought we not demonstrate that the proper integral converges on every bounded interval before asserting the existence of the improper integral?" And he'd throw chalk at me. Every time.

I still got an "A" in QM 102, because Vogt liked my paper about the positron. I was rather happy about including my interview with Carl Anderson in the paper, so I asked Herr von Vogt what he thought of that. He said, "You got an 'A' because you woke him up!" (Anderson was about 65 years old at the time, and spent most of the day sound asleep in his office.)

I still think that Wikipedia articles about mathematical subjects should strive for precision of expression. But I'm not going to make a big deal of it. I understand enough about physicists to know that in general, they just don't care about logical consistency in mathematics. So a number can be a function "by consensus", and it really doesn't matter to me. DavidCBryant 20:18, 27 January 2007 (UTC)

OK, but at Caltech they've got Math and Physics in the same division, so why the argument? I got my Caltech degree in Engineering, and skipped QM after Phys 2, so I usually avoid that level of argument. But I'm interested in trying to understand what you're saying here. Are you saying that in strict mathematical language there is not a function called the logarithm? And why? Here's another old math book that disgrees: Functions of a Complex Variable, y Edgar Jerome Townsend, 1915 Dicklyon 22:15, 27 January 2007 (UTC)

I'm copying down my comments from above, so someone will read them. First, history. At http://members.aol.com/jeff570/l.html, we find that Napier first used "logarithmus" around 1614 for a number; its English equivalent, "logarithm", occurs soon after. Contrast that with "logarithmic function", which does not appear for another 200 years.

Let's move on to a mathematical distinction, which I will illustrate with "polynomial". The expression "4x³−3x" is a polynomial, formally a member of the ring Z[x]. It is an algebraic object, like a complex number, which we can play with in various ways. One of those ways is evaluation, substituting a number for x to get a number for the expression. Using this we can define the "polynomial function" p:R→R, where p(t) equals the evaluation of the aforementioned polynomial at t. I claim that we need to be equally careful about distinguishing a "logarithm" from a "logarithmic function", at least sometimes.

And I will add a non-mathematical example, for those who still don't get it. A "telephone" is not the same thing as "calling on the telephone", even though we may say "Please telephone me", meaning the same thing as "Please call me on the telephone." Please do not bother to protest that no one will confuse the two; people learning new mathematics routinely get confused by far less! Nor are we saying that no one does, nor should, say "telephone me." Got it? --KSmrq^T 02:57, 28 January 2007 (UTC)

I didn't respond the first time, but I thought your analogy with the polynomial and polynomial function was slightly strained, since the polynomial is not a number; the cosine versus cosine function that someone mentioned is a closer analogy. Would you argue that in that case a cosine is a number and that the cosine function should not be called simply the cosine? Obviously cosine, logs, polynomials, etc. are widely used as functions, and as far as I can tell it is perfect normal (possibly not perfectly rigorous) to refer to those functions simply as the cosine, the log, etc. In other words, I don't think we have any difference of opinion or understanding about that distinction, just about what is considered normal terminology. When logarithms as numbers were invented we didn't have a well developed math of real analysis, etc., so there was little chance that the logarithm would have been conceptualized as a function at that time. But certainly it is by the nineteenth century; I've found and mentioned several books that say the logarithm is a function, without agonizing over it. I never had any great cognitive dissonance with a table of logs or a table of sines being in conflict with thinking of it as a table of values of the sine or values of the log for different arguments, which is of course how it's laid out.

So I remain a little unclear on this fundamental question: do some of you claim that it is NOT usual and customary to say "the logarithm is a function" or "a logarithm is a function"? Dicklyon 04:03, 28 January 2007 (UTC)

This is belaboring the obvious, but I'll try one more time. Grammatically, "logarithm" is a noun. A logarithm is a number, and one ought to say "logarithmic function". Consider the analogous case of "exponent". "Exponent" is a noun, an exponent is a number, and we speak of the "exponential function". Nobody says "exponent function". At least, I've never heard it.

"That's a great deal to make one word mean," Alice said in a thoughtful tone.

"When I make a word do a lot of work like that," said Humpty Dumpty, "I always pay it extra."  ;^> DavidCBryant 14:09, 29 January 2007 (UTC)

Nobody says exponent function because they are called power functions, still with an attributive noun. CMummert · talk 14:28, 29 January 2007 (UTC)

I understand that the Germanification of English continues apace. I can even parse the difference between "networkcomputersystemmaintenanceprogrammer" and "networkcomputersystemdevelopmentprogrammer" without a dictionary. That doesn't mean I have to like it – or write that way, either. DavidCBryant 16:43, 29 January 2007 (UTC)

Common usage

Rather than argue about the logic of how we think terms should be used, shouldn't we be following reliable sources and representing actual usage there? For example, to get a quick survey of sources more reliable than web pages, take a look at google book search. Search for phrases "derivative of the logarithm", "derivative of the logarithm function" and "derivative of the logarithmic function", and it becomes apparent that that former is an order of magnitude more common than the other two. It seems very clear that the word "logarithm" is used to refer to the function, and for us to assert reasons to avoid that seems un-wikipedia-like. No? Dicklyon 17:32, 29 January 2007 (UTC)

The article currently says

The word "logarithm" is often used to refer to a logarithm function itself as well as to particular values of this function.

I don't think there is any actual disagreement over the content of the article. The discussion on the talk page seems to be about grammar, usage in the real world, etc. but the comments don't seem to directly relate to any part of the article. Can anyone describe any specific changes they would like to see in the article text? CMummert · talk 17:45, 29 January 2007 (UTC)

Article mentioned in Economist's website "top story"

Just an FYI, but the top story on The Economist's website calls this article "admirable" and cites it as an example of a quality Wikipedia page. Not sure there are any special actions/notice needed... --Dfred 01:55, 11 March 2007 (UTC)

Different image (svg)

I've just uploaded a new image of the logarithmic functions (adding log0.5) from a german image used in the norwegian article. I created it in gnuplot, and it is a 640x480px SVG-image. SVG has the advantage that it is scaleable, and does not work the same way as bitmap-images. You can scale it as much as you want to, without a loss in quality. I have not altered the article, because the one that is now used, looks better (but has poor bitmap quality). The image is located here: Image:Logarithmic_functions.svg. I altered the norwegian article, located here (image in 300px): no:Logaritme. Chrtsta 21:30, 16 March 2007 (UTC)

The lead

The new leads are interesting, but where it says "x in base b" it is misleading, I think, as the base can be read as applying to x, instead of where it belongs. We still have to issue of how best to use "power". I agree with JRSpriggs that saying x is the nth (or yth) power of b is good; but it's also common to call n or y the power. What I objected to before was calling x the power without the nth that would clarify the meaning. What sources do we have that will help us choose good wording in this space? Ideas? Dicklyon 16:46, 22 April 2007 (UTC)

Woodstone, it would be best it you would not ignore JRSprigg's point, and vice versa. "Power" is used in several ways, and while you're both right, it is probably best for us to avoid takig sides. If you look in Google Book Search, you'll find "The logarithm of a number is the exponent of the power to which a given base must be raised in order to equal the number" as well as some that say "exponent or power". That is, when we say that 8 is the third power of 2, we might call 8 the power or call three the power; both are done. So instead of warring over it, admit it and find a less ambiguous way. Dicklyon 23:02, 23 April 2007 (UTC)

I have been trying to find a way to say it that uses the word "power" correctly while still being consistent with common (and often abusive) usage. But some people on the other side do not care about being correct, they just want to use the common (and very confusing) abusive language.

What are the powers of three? Are they 3, 9, 27, 81, 243, ... or are they 1, 2, 3, 4, 5, ...? I think that the answer is obvious. JRSpriggs 09:55, 26 April 2007 (UTC)

There appear to be two equally common and valid ways of talking about powers:

• "2 to the power 3 is 8" (here the "power" is 3)
• "8 is the 3rd power of 2" (here the "power" is 8)

Depending on the context the word "power" takes on a different meaning. Stating in the lead that the log is "the power to which the base must be raised", in my eyes this is hardly ambiguous. However, I'm open to ways of making it even more clear. −Woodstone 10:07, 26 April 2007 (UTC)

How do i do this on my ti-83 plus?

may someone please tell me how i would do this problem on my ti-83 plus? TIA

http://upload.wikimedia.org/math/2/2/d/22d510a23d2dc8e9cdfead05878785ba.png

ignore the 4 69.138.209.159 23:57, 25 April 2007 (UTC)

use log(81)/log(3); it doesn't matter what base the log is, it will give you log3(81). Dicklyon 00:07, 26 April 2007 (UTC)

thanks alot i appreciate the answer 69.138.209.159 21:28, 26 April 2007 (UTC)

A SIMPLER Definition of Logarithm

Could the start of the article make reference to a definition of "Logarithm" for the "lay person", and then launch into the detail and explanation of related matters? It would be really useful to know what it actually represents. —The preceding unsigned comment was added by Applet (talk • contribs) 10:11, 26 April 2007 (UTC).

Isn't that what it does? How could you get an "layer" than what's there now? Michael Hardy 22:54, 25 June 2007 (UTC)

You could get "any more lay" by taking the equations out of the first sentence and using words like "how many times you have to multiply a number times itself to get another number". I'm not sure it's a good idea or an important goal, but one could go that direction more. Dicklyon 22:57, 25 June 2007 (UTC)

I would certainly appreciate examples. All I can tell from the equations is that they appear to balance and are internally consistent symbolic structures. Beyond that, I get agita looking at it. The article says logarithms help reduce the complexity of calculations, but looking at the equations on the page makes it seem quite the opposite. General Ludd (talk) 03:11, 16 September 2008 (UTC)

I attempted to define it by re-wording the definiton; although some people claim I abused the definition of a power. Even so, I think it would be helpful to the lay person in understanding the basic concept of logarithms (that's exactly how I understood them). I said: If log base b of x = y, that also means base b to what power (b to the yth power) is x? That really set off the lightbulb for me. Hopefully this helps.Geoboe84 19:00, 26 April 2007 (UTC)

Quite often I notice that Wikipedia articles about math and science jump straight into the middle of things without ever providing a general definition of a term. All the "x to the y to the base to the yth" means nothing to a person who has no prior knowledge of the entire set of terms and concepts. I think a general definition should be a sine qua non for all entries. To create a general definition, you must do two things: 1) assume your reader is an average fifth grader; 2) include no symbols or jargon--use only plain language. If you can't come up with an initial definition or explanation that makes sense for that reader, or you've included any scientific or mathematical symbols, then you probably haven't communicated the idea. I know it's difficult to see concepts we know well from the perspective of someone with no prior knowledge, but without that perspective you may as well be writing in cuneiform. —Preceding unsigned comment added by 63.101.39.6 (talk • contribs)

Could you PLEASE sign your comments? All you have to do is put four tildes (~~~~) at the end; that add's the date and time and who posted (even if it's only an IP number, it may help us keep track of whether two comments were by the same person or different persons. Michael Hardy 22:57, 25 June 2007 (UTC)

... and where do you find these definitionless articles? This one begins with a definition that any 7th-grader would understand immediately. Mathematicians are sticklers about defining their terms, and I have many thousands of Wikipedia edits in math articles, and it looks to me as if those articles follow that sticklerhood very thoroughly. Can you cite some examples? Michael Hardy 22:57, 25 June 2007 (UTC)

This article doesn't state what a logarithm is. Oxford Concise Dictionary says: 'logarithm: one of a class of arithmetical functions tabulated to assist calculation by assisting addition and subtraction for multiplication and division, and the latter two for involution and evolution ...' That's a definition. Encyclopedias are written by specialists for laypeople.202.130.159.184 06:38, 14 September 2007 (UTC)

I think you mean substituting, not the second assisting.

I, too, would prefer to define the logarithm as a function. We launch right right into saying what the logarithm of a number is before saying the logarithm is a function, which is very confusing. On the other hand, most online definitions do it sort of like we do now. We need some good book sources; your Oxford is one such. Dicklyon 14:55, 14 September 2007 (UTC)

Re the quote above from Oxford Concise, logarithm: one of a class of arithmetical functions tabulated to assist calculation by assisting addition and subtraction for multiplication and division, and the latter two for involution and evolution ...: That's not a definition! Which class? Does it mean that any function tabulated to assist calculation is a logarithm? Does it really need to be tabulated? (I don't think logarithms to base 6.4232101 have ever been tabulated!) Does it need to be so with the purpose of assisting calculation? (I don't think logarithms to base 6.4232101 are particularly helpful!) What's involution and evolution, anyway?--Niels Ø (noe) 15:13, 14 September 2007 (UTC)

It's a partial definition. So is the one we have now. Involution and Evolution of someone-out-of-favor terms for computing roots and powers. Dicklyon 15:23, 14 September 2007 (UTC)

I too came to the discussion page looking to see if anyone was working on a simpler and clearer (i.e. more helpful for us idiots) explanation and introduction to a logarithm, without direct reference to other complex mathematical concepts. The first sentence does not explain it, it does define it. There's a difference though, and that definition might be good for a math text, but it's not as helpful in an encylcopedia for the general public. Is this concept too hard to explain in words without referencing equations? I know that is often difficult in math, but perhaps someone here would give it a shot, or reference some genius who already has. (BTW, I renamed this topic because there were already a few discussion with "definition" in the heading.)--Gatfish 00:10, 2 November 2007 (UTC)

This is not a textbook, and an informal definition in plain words will do no harm. No need of functions for that - I guess Napier and the other old boys did not know about functions, and I'm sure my father - a carpenter who was taught the use of log tables for multiplication etc. - didn't. So I've tried my hand with a new first sentence and a simple example following it. Perhaps this new example and the example ending the lead should be harmonized somehow.--Niels Ø (noe) 09:57, 2 November 2007 (UTC)

Equivalent to

I'm probably just being picky but I don't like that the intro says

y = $\log _{b}(x)\,\!$

is equivalent to

$x=b^{y}\,\!$

I have always used equivalent to when talking about, for exmaple, trigonometric identities - using it in this case just seems incorrect somehow and a misuse of mathematical symbols. I suggest it be reworded. Algebra man 11:20, 14 July 2007 (UTC)

I don't get it. Do you want to avoid "equivalent to" because you associate it with trigonometric identities? —Bromskloss 12:11, 14 July 2007 (UTC)

No. I suppose it wasn;t clear; to me it sounds like we are sayin y=x (the trig thing was just saying that for, in say tan(a+b), both sides will always equal each other egardless of the values the two variables take - it's not really relevant, I shouldnt have brought it up). Algebra man 12:13, 14 July 2007 (UTC)

"Equivalent to" is a very frequently used and universally standard phrase in mathematics. To some extent its meaning varies with context, but by default "A is equivalent to B" means "A is true if and only if B is true". I have long been accustomed to thinking that everyone who's finished high school knows this. Consequently I was surprise when, while explaining trigonometry in the classroom, a student asked "By 'equivalent to', do you mean 'equal to'?". I did not mean "equal to". I said the identity we were trying to prove is readily seen to be equivalent to another identity. Certainly "equal to" would have been horribly inappropriate, since in the second identity, equivalent to the first, the two expressions were not equal to those in the first identity. "Algebra man", if you're going to be writing about trigonometry, you should know that, since it is standard usage. I will certainly oppose your proposal. The equality

$y=\log _{b}(x)\,\!$

is indeed equivalent to

$x=b^{y}\,\!$

and it is altogether appropriate that the article should say so. Michael Hardy 13:22, 14 July 2007 (UTC)

Alright, keep calm, I only suggested it - if it's correct then I agree it should be kept. Algebra man 13:38, 14 July 2007 (UTC)

Log base two code example

I didn't change anything on the code example, because I know I'm wrong a lot of the time :)

I can't, however, see how the explanation, or the code, are true or accurate representations of any way of computing a log of base two. [EDIT -- I Forgot this bit:] The integer part of the logarithm to base 2 of an integer is given by the position of the left-most bit, and can be very quickly computed using the following algorithm: [/EDIT 69.65.232.61 00:00, 20 September 2007 (UTC)]

 int log2(int x){
   int r = 0;
   while( (x >> r) != 0){
     r++;
   }
   return r;
 }

Okay, so let's say our x we pass to this function is 16, or 0b10000. According to the definition of the "quick way" of doing logs of base two, the integer part of our answer is simply the position of the left-most bit of our number passed to the function.

You can see right away that our answer, then, would be 5. To be consistent with the example code, let's work through it.

 We start with x = 0b10000
 We start with r = 0.
 Check if shifting 0b10000 right by 0 is 0 -- it's not. It's 0b10000 Continue with our while statement.
 Increment r by 1 to make it 1.
 
 Is 0b10000 shifted to the right by 1 equal to 0? It's not. It's 0b1000. Continue.
 Increment r by 1 to make it 2.
 
 Is 0b10000 shifted to the right by 2 equal to 0? It's not. It's 0b100. Continue.
 Increment r by 1 to make it 3.
 
 Is 0b10000 shifted to the right by 3 equal to 0? It's not. It's 0b10. Continue.
 Increment r by 1 to make it 4.
 
 Is 0b10000 shifted to the right by 4 equal to 0? It's not. It's 0b1. Continue
 Increment r by 1 to make it 5.
 
 Is 0b10000 shifted to the right by 5 equal to 0? It is! Break out of our while loop.
 
 Return the answer that log base two of 16 is equal to 5.

Now, what's two to the fifth power? 32. 32 is not equal to 16.

This example fails to recognize that the rightmost bit is 2^0. If we pass 0 to this function, we'll get the answer that log base two of zero is zero. That's not right, but let's go ahead and leave out error catching, and assume that the function can always handle it's inputs, i.e., they're assumed always to be greater than or equal to zero (Since our function finds the integer part of the log base two of our input, any floating point numbers passed will [should?] be truncated by the data type, and the while loop will be skipped, returning an integer part of 0). If we pass 1 to this function, we'll get the answer of 1. That's off by 1. Just like the answer 5 is off by 1. The quick fixes are to either return r-1, or change our while statement to, instead of "not zero," "(x >> r) > 1". I'm sure there are more ways to fix this, but I'm not any kind of computer science whiz, so there might be more elegant ways of fixing things.

That's my understanding of this whole example. I could certainly be way off... And just be embarassing myself. :( -- 69.65.232.61 23:58, 19 September 2007 (UTC)

The definition is correct (the rightmost bit is bit 0). The implementation is wrong. It's unclear what to do about the 0-case, but I suppose returning "-1" for error is okay. The algorithm also fails for negative numbers. ⇌Elektron 12:45, 22 September 2007 (UTC)

I fixed the code, including a short comment about the affect or returning r-1 instead of r. The log base 2 of 16 is 4, not 5, which makes sense if count bit positions starting with 0 on the right. Dicklyon 16:28, 22 September 2007 (UTC)

Relevance of MIDI

The fact that "middle C" is 60 in MIDI is about as relevant as the fact that it's 40 (I think) on an 88-key grand piano, if not less relevant. For the most part, it's used as a unique identifier for the note — it doesn't matter if it's in order, or a random order, or a grey-code order, since most instruments will use a look-up table to pick the right waveform anyway (whereas on a real piano, it does matter, since it's used to check that the keys are in the correct order). I don't know of any software which reports the MIDI note number in preference to the note name, and for some instruments (like a drum kit) it has absolutely nothing to do with frequency.
Pitch bends, on the other hand, are integers from -8192 to 8191, corresponding to -1 semitone and +1 semitone (almost, I suspect). If this was combined with the note number, then it might be worth mentioning, but it's not. The note number does exactly what it says: it numbers notes in an arbitrary order. It doesn't prescribe frequency (I suspect you can get well-tempered piano samples somewhere), and we certainly don't have fractional note numbers, which the original phrasing seems to suggest. It may be relevant to interval (music), but I doubt it's relevant to Logarithm. ⇌Elektron 15:30, 23 September 2007 (UTC)

You seem to be unaware of the functions in MIDI for microtuning. If it is desired to play in another tuning than equal tempered, it is possible to indicate very precisely the frequency of the required notes. This is accomplished by defining a logarithmic scale filling in the ranges between the semitones of the equal tempered scale in a compatible way. This scale corresponds to the note numbers for whole semitones. Please read up before you delete. You can start at MIDI Tuning Standard. −Woodstone 15:46, 23 September 2007 (UTC)

I looked it up and found that midi uses both note numbers and interval microtuning in cents. So there's something worth saying about all that, but what was there was incorrect, partly due to my error in editing. Maybe I'll work on it later. Dicklyon 15:48, 23 September 2007 (UTC)

The bit about MIDI is now sandwiched between the mention of fractional semitones and the definition of cents, making it look like cents is a MIDI-specific concept. In fact, it dates at least from Helmholtz, On the Sensation of Tone (1885) and Broadhouse, Musical Acoustics... (1881). Dicklyon 19:09, 23 September 2007 (UTC)

Of course cents predate MIDI. If the current formulation suggests otherwise, that was unitentional. −Woodstone 19:26, 23 September 2007 (UTC)

I figured it was probably unintentional, but since my fix got undone by yours, you should work on it. Dicklyon 19:30, 23 September 2007 (UTC)

Your latest mod omits the fact that midi fill in using cents. Dicklyon 19:32, 23 September 2007 (UTC)

Actually MIDI does not really use cents. Some of the commands fill the space of one semitone in small steps, but these are always a power of 2 in number (e.g. 8,192=2¹³ steps of about 0.012207 cents each), never a 100 steps. −Woodstone 21:16, 23 September 2007 (UTC)

New old definition images

I would question the wisdom of including a full shot of a page containing another definition of the logarithm (I mean the second page picture, not the first). If there's something useful from there to merge in the article, that would be nice. Otherwise, the purpose of illustrations is to complement the text with some visual description rather than providing alternative definitions. Oleg Alexandrov (talk) 04:56, 12 February 2008 (UTC)

I'm not so attached to the picture, but I think it is a useful mechanism for showing an alternative development that gets straight through a simple definition and on to the usefulness of logarithms. This might be useful for people who have a hard time following our slight mroe mathematical approach. And it's only a small fraction of a page image. What do others think? Dicklyon (talk) 16:22, 12 February 2008 (UTC)

Should they be on top though, before the logarithm graph? It is much more important to have pictures at the top of the page than yet more defintions I think. Oleg Alexandrov (talk) 03:27, 14 February 2008 (UTC)

I like the definition pictures a lot, and would keep both if it were my article, but I think the graph should be at the top. CRGreathouse (t | c) 04:46, 14 February 2008 (UTC)

I think the graph is more important, too. Alternative definitions should (if at all) not be given in some scanned old encyclopedia, but in the regular text (perhaps in the history section). Jakob.scholbach (talk) 08:35, 14 February 2008 (UTC)

I've move the graph to the top and the second definition down to the history section. Maybe the first def should be moved down there as well. The images are useful as they document the history of the idea but they have more context close to where the history is discussed. --Salix alba (talk) 09:59, 14 February 2008 (UTC)

I find it strange to take a picture of text - if we want to discuss that old text, we should be able to quote it, rather than take a picture of if. It's not as if we're trying to comment on the typeface or typesetting of the text, only the mathematical content. — Carl (CBM · talk) 12:21, 14 February 2008 (UTC)

While quoting is often a great alternative, it might not be so great here as a way of including alternative definitions, as that just gets kind of dry and repetitive. I find that images of old pages are a great way to frame a historical viewpoint (I've done similarly in quite a few other articles); the old font and old look of the page make it clear that this is not something modern, and that the idea goes way back, even if differently conceptualized. In fact, now that I think of it, we need a close-up of a log table to illustrate how they were used; that would be way better than making a new log table in wiki formatting, seems to me. Dicklyon (talk) 03:21, 15 February 2008 (UTC)

Beaufort

In the article on the Beaufort wind force scale, there is the explicit formula:

v = 0.8365 B^3/2 m/s

with v the wind velocity and B the Beaufort scale number. This gives an excellent approximation of the scale and shows that it is essentially a logarithmic scale.−Woodstone (talk) 15:06, 20 February 2008 (UTC)

Notice that none of the variables are in the exponent. This is called a power law, not an exponential. So I took it out. Dicklyon (talk) 16:11, 20 February 2008 (UTC)

You are right. My mistake. I should have paid more attention. −Woodstone (talk) 16:15, 20 February 2008 (UTC)

Introduction

Let's have the most useful information first, please. The first paragraph is overly complex and confusing. I'm not too familiar with line spacing in this Wiki; it would be nice to even things out a little bit: lowering "4)", for example. Thanks.

In mathematics, the logarithm of x to the base a is written as log_ax and is defined by the statement:

$If~x=a^{y},~~then,~y=log_{a}x$

The logarithm is the inverse operation of exponentiation and obeys four laws:

$For~n,~x,~y\in \mathbb {R} ,~~x>0~and~y>0,$

$1)~~log_{a}(xy)~=~log_{a}x~+~log_{a}y$

$2)~~log_{a}\left({x \over y}\right)~=~log_{a}x~-~log_{a}y$

$3)~~log_{a}x^{n}~=~nlog_{a}x$

$4)~~log_{a}x~=~log_{a}y~~if~and~only~if~x=y$ --Charlesrkiss (talk) 05:31, 2 March 2008 (UTC)

Just don't use more than one blank line between things, and settle for the spacing you get. I don't understand your complaint. The lead goes to great lengths to express what a log is in the simplest possible language, rather than implicitly via a bunch of equation properties; this was done in response to requests to make it more approachable by the non mathematically facile reader. Does it mess you up to have all those words instead of equations? Do you think what you've shown above would be a better lead? I don't. Dicklyon (talk) 05:39, 2 March 2008 (UTC)

For example,

$1)~~log_{b}(x)=log_{b}(b^{n})=n.$

Too much b, is what I think; one must imagine variable b as a coefficient and a base. That's a bit advanced. Besides, most math books I've seen make it clear enough with:

$2)~~If~x=a^{y},~~then,~y=log_{a}x$

IMHO What could be more clear? --Charlesrkiss (talk) 06:02, 2 March 2008 (UTC)

Another criticism is this: directly above, equation 1) has three b's, two n's and one x; while equation 2) has two a's, two x's and two y's. If one believes in "mapping" as an intuitively simple process of pairing one entity with another, then, because equation 2) is a set of three pairs, it should be simpler than Equation 1) which contains an odd and complex distribution of entities; seems like trying to follow the bouncing ball in a game of table tennis, I think. --Charlesrkiss (talk) 06:37, 2 March 2008 (UTC)

Besides, if I need to be reminded of the rules of "logarithm" and it's definition, I should be able to click Wikipedia, search "logarithm", and find the information at the top of the page. The article contains this information in weird formats, in different contexts, spread throughout the page; chaotic. —Preceding unsigned comment added by Charlesrkiss (talk • contribs) 17:24, 2 March 2008 (UTC)

It seems to me that the essentials of the logarithm are:

${\mbox{if}}~~x=b^{y}~~{\mbox{then}}~~y=\log _{b}x\,$

$\log _{b}(xy)=\log _{b}x+\log _{b}y\,$

The others above (powers, quotients, unicity) are direct consequences

−Woodstone (talk) 21:32, 2 March 2008 (UTC)

It's even simpler than that: The first one alone captures the essential idea; all else follows logically from that. Michael Hardy (talk) 21:40, 2 March 2008 (UTC)

You are right of course, but I would see the first as the definition and the second as the primary consequential utility. I have meanwhile simplified the lead a little. −Woodstone (talk) 21:43, 2 March 2008 (UTC)

"the essential idea?" Seriously, why not just have the definition of logarithm, and leave the rest of the page blank, then? I thought the purpose of this page, and Wikipedia in general, was to saturate completely the context of the subject matter, including all subjects!

Therefore, I would prefer the definition, followed by it's known consequences with proofs. Applications. Historical information. Links. And so on. More or less, in that order!--Charlesrkiss (talk) 07:15, 16 March 2008 (UTC)

Value of the logarithm

Historically the logarithm was important in practical calculation, log tables, slide rules and all that. This utility has gone by the wayside today with calculators. The real utility of the logarithm is in its mathematical applications. It seems to me that the novice reader would be justified in asking: "What are logs good for?" The article provides only an historical value. I hesitate to involve myself in an interminable editing war over this matter. Brews ohare (talk) 16:36, 4 March 2008 (UTC)

In order for the reader to have an understanding of logarithms in historical context, like why you still finds books of nothing but tabulated logarithms in used book shops, and why high-school students had to learn to use them, you have to treat them up front as primarily aids to computation. The analytical function interpretation came much later, and is of primary interest to the mathematician. The lead needs to acknowledge both aspects of a what a logarithm is. Dicklyon (talk) 17:51, 4 March 2008 (UTC)

Historical context interests a subset of readers. So does the "analytical function". I'd venture a guess that neither group is large. Brews ohare (talk) 18:24, 4 March 2008 (UTC)

Logarithm tables are hardly just a historical curiosity. In fact they're still very much in use, except by computers rather than humans. On many processors a division takes about as much longer to perform than an addition as it does to most humans, making an approximation by way of logarithm and power tables a useful alternative. Of course todays PCs have fast floating point units and severely limited memory bandwidth, but things are quite different in embedded systems. Perhaps this ought to be mentioned somewhere in the article? --83.166.6.212 (talk) 08:21, 11 April 2008 (UTC)

Simple?

$\log _{b}(x)={\frac {\log _{k}(x)}{\log _{k}(b)}}:$ simple: $\log _{a}(b)={\frac {\log _{c}(b)}{\log _{c}(a)}}.$ —Preceding unsigned comment added by 150.217.1.25 (talk) 15:07, 17 March 2008 (UTC)

Both say the same thing. Nschoem 19:48, 14 May 2008 (UTC)

Limeric on additivity of logarithms

To bear offspring Noah's snakes were unable;
Their fertility was somewhat unstable.
So he built them a bed
out of tree trunks and said,
"Even adders can multiply on a log table."
199.196.144.13 (talk) 14:44, 2 June 2008 (UTC)

Properties Error?

This might be minor, and I might be wrong, but I believe that $\left(b^{\log _{b}(c)}\right)^{p}$ should have "b" outside the parentheses with ^p inside of them, if you really want to break exponentiation down into steps. These are not equivalent, nor do they strictly follow the log power rule. —Preceding unsigned comment added by 71.249.62.225 (talk) 03:21, 8 September 2008 (UTC)

No, it is correct. For example, log₂(8)=3, and

$(2^{3})^{2}=2^{6}=64=8^{2}$

but

$2^{(3^{2})}=2^{9}=512\neq 8^{2}$

Gandalf61 (talk) 09:12, 8 September 2008 (UTC)

log(a+b)

Nice article, but it does not answer what is the outcome of log(a+b)=? —Preceding unsigned comment added by 195.128.2.68 (talk) 21:59, 18 October 2008 (UTC)

More a question for the reference desk. The operation is most of use as far as I know in embedded arithmetic as the log number system. You might be interested in for instance Method to Approximate the Logarithm of a Sum Dmcq (talk) 23:34, 18 October 2008 (UTC)

It's not a question that has a simple or particularly useful answer. But it is sometimes expressed, for a and b such that a is positive and greater than the absolute value of b. Writing a + b as a * (1 + b/a), the log of a + b is log(a) + log(1 + b/a); the latter term is sometimes usefully approximated, e.g. as a power series or a table lookup, since it has a limited domain. Dicklyon (talk) 04:49, 19 October 2008 (UTC)

This formula is extremely useful for calculation of logarithms. With a table of precomputed values for log(a), or repeated application with a = 2^n it allows one to get arbitrarily rapid convergence in the Taylor series for log, which otherwise converges very slowly or not at all. Or if one is only interested in low precision, it gives the accurate approximation log(a+b) ≈ log(a) + b/a for |b| < |a|. There is also an equivalent formula for atan, atan(x+y) = atan(x) + atan(y/(x^2+xy+1)), that should be mentioned somewhere. Fredrik Johansson 09:04, 19 October 2008 (UTC)

You might also like "Comparing Floating-point and Logarithmic Number Representations for Reconfigurable Acceleration referenced from floating point where they got big speed and space gains by using this operation for the adds and just adding for the multiplies. As an aside I managed to cram a large number of approximate counters in a small space once by only incrementing them with a probability that went down the bigger the count so I got an approximate log count in the counter Dmcq (talk) 11:21, 19 October 2008 (UTC)

logarithm of a negative or complex number

log e to the 2pi x i = log e to zero power this section is confusing as it assumes reader's knowledge of what e represents i suggest hyperlink e and i to their respective definition pages Hellacious abuser (talk) 13:51, 30 May 2009 (UTC)

Did you read the earlier reference to complex numbers? It is pretty difficult to read anything about complex numbers without meeting 'i and e. I would have thought these were prerequisites to have got that far. Dmcq (talk) 15:48, 30 May 2009 (UTC)

I wlinked it in another way, instead of a diverting footnote. But also I think anyone who bothers to read about complex logarithms, doesn't really to be explained i, e or ln. --J. Sketter (talk) 09:22, 8 October 2009 (UTC)

New error, or old?

How long has this gone unnoticed? "Also, the logarithm of 0.693..., using base e, is 2, since 0.693... = e2." AstroLad47 (talk) 05:05, 15 July 2009 (UTC) AstroLad47

New. Dicklyon (talk) 05:26, 15 July 2009 (UTC)

Why logarithms?

"The human eye responds to changes in light intensity on a logarithmic scale." - http://www.astro.northwestern.edu/labs/m100/logs.html
Is this sort of practical application of logs mentioned anywhere on wikipedia yet? Logs are cool. Just added this quote to high dynamic range imaging.

Unfortunately, it's a common but untrue misconception. See Weber–Fechner law and Stevens' power law. Dicklyon (talk) 15:39, 20 August 2009 (UTC)

Hi.

"The eye senses brightness approximately logarithmically over a fairly broad range." - Weber–Fechner_law#The_case_of_vision.

"Other researchers would now regard the power law as refuted and only of historical interest (see criticisms below)." - Stevens'_power_law. "Criticisms" is the largest section of prose on that page.

--Darxus (talk) 04:07, 21 August 2009 (UTC)

Oh. Stevens' power law is logarithmic too. My math is not so good, but the whole use of exponents is the giveaway. --Darxus (talk) 04:13, 21 August 2009 (UTC)

No, having exponents is not enough. It kind of depends on what's in the exponent. A power law is never logarithmic. Logs behave pathologically as the physical stimulus approaches zero. Power laws have a slope singularity there, but a finite (zero) value, which makes them quite a bit closer to what physical systems can actually do; power laws aren't perfect either, but the fit better over a wider range than logs do, and can be made to look log-like over a fairly wide range if that's what's needed to fit the data. Dicklyon (talk) 06:09, 21 August 2009 (UTC)

The criticisms are fair, but none of them suggest that the power law is not an improvement over the old logarithmic Weber–Fechner law, which is subject to all the same criticisms and more. Dicklyon (talk) 06:22, 21 August 2009 (UTC)

Nobody has mentioned "DECIBEL" in this entire article. I thought that decibel which uses logs was the MAJOR "killer app" for this. We don't use slide rules anymore. Baruchatta (talk) 14:55, 17 November 2009 (UTC)

Actually it's mentioned early in [[8]].

Logarithms and Inequalities

I may just be a math newbie, but I couldn't find any reference to this online or in my brain.
I want to solve an inequality of the form $a<b^{n}$ for $n$
So, I take $log_{b}$ of both sides to obtain $log_{b}(a)<n$.
This is correct for b \geq 1</math>, but incorrect otherwise.
Specifially for $0<b<1$, since logarithms with negative base are complex and I don't want to get all messy dealing with complex math.
For example, take $a={\frac {1}{8}}$ and $b={\frac {1}{2}}$.
It is obvious, if ${\frac {1}{8}}<({\frac {1}{2}})^{n}$, that $n<3$.
But, if we solve the inequality as if we cannot see that, we end up as follows:
$log_{\frac {1}{2}}({\frac {1}{8}})<log_{\frac {1}{2}}(({\frac {1}{2}})^{n})$
$log_{\frac {1}{2}}({\frac {1}{8}})<n$
Evaluating the LHS gives: $n>3$
But, we know $n<3$ so, this result is incorrect. The inequality should have flipped.
So, to fix this, I am adding a note on the page to deal with logarithms and inequalities that should read something like this:

When taking the logarithm of both sides of an inequality such as $a<b^{n}$, if $0<b<1$ then you must flip the inequality.
So, when $0<b<1$, $a<b^{n}$ becomes $log_{b}(a)>n$.

P_iK^o 18:35, 3 September 2009 (UTC)

Proof for existence

I want to introduce in the article the proof for the existence of the logarithm in the real case(mainly from first principles). This proof can be included in the section of "Logarithm as a function" or in a new section altogether. It can even be inserted using a hide/show option. The proof runs as follows:

Fix b > 1, y > 0. Then there is a unique real x such that b^x=y. This x is called the logarithm of y to the base b. The proof is divided into various steps.

(a) First note that for any positive integer n, bⁿ-1≥n(b - 1). This is true as each of b^n-1, b^n-2,...b are greater then 1 and summing and then applying the forumla of the sum of a finite geometric series proves the claim.

(b) The second step involves proving that b - 1 ≥ n(b^1/n - 1). Note that as b^1/n > 1 so by (a), (b^1/n)ⁿ - 1 ≥ n(b^1/n - 1).

(c) The next step is showing that if t > 1 and n > (b - 1)/(t - 1) then b^1/n < t. This follows from b^1/n = (b^1/n - 1) + 1 ≤ (b - 1)/n + 1 < t if we use the claim established in (b).

(d) The next step is showing that if w is such that b^w < y then b^w+(1/n) < y for sufficiently large n. To see this note that 1 < b^-wy = t (say). Choose n > (b - 1)/(t - 1), then by (c), b^1/n < b^-wy or b^{w + (1/n)} < y for sufficiently large n.

(e) Similar to (d) it can be now noted that b^w > y, then b^w-(1/n) > y for sufficiently large n.

(f) The following claim can now be established: Let A be the set of all w such that b^w < y. Then x = sup A satisfies b^x = y.

Proof: From (a), bⁿ ≥ n(b - 1) + 1 for all n. For which each z in R choose an n so that n(b - 1) > z - 1 or n(b - 1) + 1 > z. Hence for all z we have an n such that bⁿ ≥ n(b - 1) + 1 > z. Hence the set {bⁿ : n ∈ N} is unbounded. Now consider the function f : R → R defined by f(x) = b^x. Clearly f is an increasing function. This follows from the definition of b^x. Note: Rigorously b^x can be defined as follows: First define bⁿ for positive integers n in the obvious way, i.e. b multiplied to itself n times. Extend this definition to all integers by letting b^-n (n > 0) to be 1/bⁿ, and b⁰ = 1. Now extend this definition to all rationals by letting b^m/n as (b^m)^1/n. Finally extend the definition to all the reals by defining B(x) = {b^t: t ∈ Q, t ≤ x} and letting b^x = sup B(x). Now clearly if x < y then as B(x) ⊆ B(y) so b^x < b^y; i.e. f is an increasing function.

Define A = {w : b^w < y}. The set {bⁿ : n ∈ N} being unbounded gaurantees the existence of a n such that bⁿ > y. Thus n is an upper bound for A. Let x = sup A.

Suppose b^x < y. By (d), for sufficiently large n, b^{x + (1/n)} < y, i.e. x + 1/n is in A. But this is impossible as x = sup A. So b^x < y is not possible. Suppose b^x > y. By (e), for sufficiently large n, b^{x - (1/n)} > y, i.e. x - 1/n is not in A. Since x - 1/n cannot possibly be the sup of A so there is a w in A such that x - 1/n < w ≤ x. But then as f was increasing, b^{x - 1/n} < b^w < y, a contradiction as b^{x - (1/n)} > y. So b^x > y is not possible.

(g) The fact that this x is unique follows from the fact the function f described in (f) is increasing and hence 1-1.

Should I go ahead and add the proof. Regards--Shahab (talk) 19:16, 12 September 2009 (UTC)

Proofs aren't normally put in unless they are notable or have some intrinsic interest or are very short. Also you didn't give a citation. So overall I'd say no in this case. Dmcq (talk) 20:05, 12 September 2009 (UTC)