Talk:Logarithm/Archive 5

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unhappy with recent edits

In recent edits, Alsosaid1987 introduced a number of "definitions" of the logarithm. I am seriously unhappy with these edits: these "definitions" are, IMO, (more or less elementary) analytic properties of the log function and should be stated as such (or omitted, depending on the relevance). While it is correct that the notion of exponentiation to real exponents requires elementary calculus, more specifically the concept of a continuous function, these "definitions" are way more complicated (as one sees from the fact that one needs integration, which in particular subsumes continuity in its definitions and proofs of existence etc.)

I suggest to completely remove these "definitions" at this stage. They can be briefly mentioned in the later parts (and indeed the first two already are, the third is not and I don't think it is particularly relevant).

I am also, somewhat less though, unhappy with edits of Purgy Purgatorio: the term b^x does not lack a rigorous definition in whatever generality. It has a completely rigorous definition, which should be the content of our article on exponentiation. (But which should not be the main focus of this introductory section here.) Comments? Jakob.scholbach (talk) 21:41, 30 April 2018 (UTC)

I believe that mathematically/historically it is correct. That doesn't mean that it can't be improved, though. As well as I know it, and maybe not well enough, irrational exponents didn't come until after calculus, and also that ln(x) was first defined through the integral, that is, area under the curve, of 1/x. Much of they way math is taught now is historically wrong, but easier to teach and learn. Among others, integration is more fundamental, as the area under a curve, than the derivative as slope. There are some calculus books that teach integration first, and derivative as the inverse operation later. (Best for students that already know some calculus.) Gah4 (talk) 22:22, 30 April 2018 (UTC)

The focus of this article, as almost any other math article should be on keeping simple things simple, and not on keeping the historical order. (We do have a history section, which has this latter priority.) Mathematical history has nearly always seen a trend to simplifying things, and we don't do the reader any service by presenting the partial / unstructured understandings present in old-day-math. Jakob.scholbach (talk) 21:09, 1 May 2018 (UTC)

In a first step I revised my statement about a lacking definition, to better express my, and hopefully also Alsosaid1987's, original intention. I badly edited his statement, which, I suppose, was intended to motivate the extensions from integer-exponentiation to rational- and finally to real-exponentiation in this ad hoc manner. As I expressed in my edit summary I am not fully convinced about having this section in even this sketchy full g(l)ory, but I certainly do plead for an explicit hint to the difficulties, hidden behind the immediate intuition presented in the beginning of the Definition section.

As regards coining definitions as properties and vice versa, I take a very flexible approach: it depends on which is which, each one taken as definition turns the others to properties (requiring different proofs, of course, and isn't the integral mentioned as the most favored, somewhere?). Considering the average level of knowledge assumed for possible target readers I think it is appropriate to offer this overview of possible definitions exactly in a section Definitions.

In any case, improvements are always possible. Purgy (talk) 06:58, 1 May 2018 (UTC)

Yes, improvements are always possible, but this section and the article as a whole has not improved, in my opinion, by the addition of this material. And I hardly see how to fine-tune these edits so that they do improve the article. Please also keep in mind that this article is a featured article, and has seen a tremendeously detailed FA review (see the logs linked above).

I think the widely agreed structure in math articles is to begin with simple things, and gradually increase depth and width of the article, so to speak. Once again, what is the merit of presenting a "definition" which can not be understood / appreciated without solid foundations in calculus if there is one which avoids this problem nearly completely?

Also, what you are (as far as I understand) attempting to explain here, is not the topic of this article, it is the topic of exponentiation!

I also disagree with the statement that defining logarithms via integrals is a "definition", let alone the most favored one. The fine point that b^x for irrational x needs explanation will not worry 99% of our readers. The remaining 1% should get a short hint "go to exponentiation if you worry / want to know about this". Attempting to create understanding of this 1% by introducing more advanced statements about logs at this point is fruitless, I am firmly convinced. Jakob.scholbach (talk) 21:09, 1 May 2018 (UTC)

The given "definition" is in principle not understanable wrt reals, but nevertheless given, the others have a chance, at least. It is true that the troubles wrt exponentiation occupy the largest place, but the log is at the end. Please, also see the outdented reply below. Purgy (talk) 11:21, 2 May 2018 (UTC)

my $0.02: My original point was to give a hint as to how much work would need to go into rigorously defining log as the inverse of exp (defined without series), to motivate the development of the standard calculus definitions as a convenient alternative (even if all the tools of calculus need to be developed beforehand). I commend Purgy Purgatorio's efforts to make things more explicit and readable. However, I understand the criticisms, and looking back, I do think these details are a bit too much, too early.

However, the previous situation was untenable, and these edits are an improvement, even if the location of these details is debatable. I find it to be disingenuous and patronizing to give a "definition" of the exponential function for integer exponents and "define" the logarithm as the inverse of this function while shoving the problems of defining b^x for real x under the rug without a mention. While I agree with Jakob.scholbach's point that most readers won't appreciate the need to have a rigorous definition, Wikipedia needs to give a definition that is correct and accurate.

I advocate the integral definition as the cleanest one. Obviously, it requires the Riemann integral to be defined (i.e., a linked page), but the point is, there is no clean and simple definition of the logarithm. High school texts routinely "define" log as the inverse of an undefined real exponential function, which is essentially circular. Wikipedia should not repeat the lies (or fairy tales) of schoolbooks. For alternative definitions, I think the prudent thing to do is to move these details later on in the article. While using the inverse definition is okay for motivation, a note should be made that defining the log as the inverse of the exponential function requires a definition of the exponential on the reals and showing that the function is in fact invertible.

I do note that analytical properties of b^x (like continuity and differentiability) are *not* needed to show existence and uniqueness of the real logarithm (see Rudin, p.22, problems 6 and 7). Only the l.u.b. property of the reals and a definition of the supremum of a set are needed. Alsosaid1987 (talk) 02:35, 2 May 2018 (UTC)

Especially for honoring the argument that this is a FA, I gave it a bold shot and shifted the subtleties section from its consensually inappropriate place down, shortened and reverbalized the "definitions"-section, and only slightly modified the header of the original Definition section. I hope this is a step towards a consensus, and I was not too bold. Maybe it is even possible to improve the article from this state without dumping all efforts.

I want to add that I agree with Alsosaid1987 on deprecating the intransparent application of the Main Theorem of Highschool Math (everything OK with rationals fits with reals) and all the other lies, fairy tales, and mystifications in math education, and that I also agree on math's history being one of simplification. I do not, however, fully understand the reservations wrt multiple "definitions" to select from to momentary predilections. I took the leap of faith to ignore all constructivists' caveats and believe in reals, but I am rather agnostic to prefer logs to exps, or vice versa. I think an encyclopedia, accepting the knowledge about a specifically wrong content is untenable, even when less than 1% of its readers is concerned. Purgy (talk) 11:21, 2 May 2018 (UTC)

I have removed any subtleties by putting the integral definition into a formal definition section. We're only talking about the natural logarithm and changes of base applied to it in this article. We are not concerned about all the silliness and tricks in the exponentiation article to deal with integers and rationals etc, this is the inverse of the exponential function. Dmcq (talk) 12:34, 2 May 2018 (UTC)

I support these changes. Will add a short sentence connecting the "conceptual definition" to the formal one. Alsosaid1987 (talk) 14:15, 2 May 2018 (UTC)

This is looking good to me, too. It needed a seque between the conceptual, which didn't mention natural or e or exp, and the formal that was all about that, so I added a sentence about that into the conceptual section. Dicklyon (talk) 14:40, 2 May 2018 (UTC)

In the lead we have:

...the defining relation between exponentiation and logarithm is:

$\log _{b}(x)=y\quad$ exactly if $\quad b^{y}=x.$

Would this make more sense in the other direction?

$b^{y}=x\quad$ exactly if $\quad \log _{b}(x)=y.$

since the implication is more generally true for complex logs? Dicklyon (talk) 14:44, 2 May 2018 (UTC)

'exactly if' seems to be another way of writing if and only if. It could probably be simplified to just have an if with the way you put it. Dmcq (talk) 14:49, 2 May 2018 (UTC)

Several people have repeatedly asserted that defining (in high-brow language) log as the inverse of b^x is not defined, that something is "specifically wrong".

I strongly disagree with this conception. The function b^x is perfectly well-defined, for example characterized (again using high-brow language) as the unique continuous extension of the ovbious function from Q to R. The arguments needed to make this precise involve basically nothing than the notion of a continuous function. In the context of this article here, it is particularly simple to explain: for irrational x, b^x can be "computed" by taking b^{some increasingly good (e.g., decimal) approximation of x}. All this is completely accessible to a 16 year old person, say. The integrals people are offering here are not accessible to those, and if you would carry it out to the end (involving log_b for b \ne e), ln x for x < 1 (talking about orientation involved in the integral)) you would see how painful this approach is.

To conclude, while it is possible to define logs using an integral, it is not helpful to the quasi-totality of our readers (for example, I personally learnt about logs 3 years before I learned about integrals). And, I insist on this point, asserting that something is lacking a definition etc. is simply wrong.

If you are commenting on this, please specifically address the benefits and drawbacks of either approach. Jakob.scholbach (talk) 20:24, 2 May 2018 (UTC)

User:Jakob.scholbach, I agree with you on that definition. However, that was not how the article was written. That's what I take issue with. The was a definition for the integers and magically the log is defined as the inverse.

Alsosaid1987 (talk) 21:43, 2 May 2018 (UTC)

You are right of course but defining logarithms in terms of the integral is the way it is commonly done in elementary calculus courses, and we should follow what is out there. Doing it the way you said is done in the conceptual definition section just before the formal definition section. The exponentation article has a whole load of special cases for integers and rationals, dragging logarithms down to the same level in formal terms is unnecessary and would just cause problems. I think we're better off having the informal bit talk like you say but the nasty mess that using it would entail for people who want things all tied down is I think better avoided and we just talk about real numbers. Dmcq (talk) 20:58, 2 May 2018 (UTC)

Re: unhappy with recent edits

I agree that User:Alsosaid1987 has been causing trouble in this article. I reverted to User:David Eppstein's version because it seems to me that everyone was trying to clean up User:Alsosaid1987's work. It appears to me that User:Alsosaid1987 was overly enthusiastic about improving this article and decided to make major additions to the article before he actually read the article first. Thank you to User:Alsosaid1987 for your efforts, but please be sure to read the article before you edit it. Brian Everlasting (talk) 15:58, 2 May 2018 (UTC)

Please read the support for these changes in the section above before accusing me of causing trouble. The change have been re-applied. In case you haven't noticed, I'm not the only one making these changes.

Alsosaid1987 (talk) 16:01, 2 May 2018 (UTC)

Let me elaborate about why I think User:Alsosaid1987 has not read this article before he edited it. According to User:David Eppstein's version, the natural logarithm is defined as:

$\ln(t)=\int _{1}^{t}{\frac {1}{x}}\,dx.$

then User:Alsosaid1987 added a definition of natural logarithm farther up the article stating:

$\log _{e}x:=\int _{1}^{x}{\frac {dt}{t}}.$

these two definitions are nearly identical, but I prefer User:David Eppstein's version. Brian Everlasting (talk) 16:11, 2 May 2018 (UTC)

Thanks for the credit, but I am merely the first of several editors to deal with Alsosaid1987's changes (in my case, by reverting them). I agree with you in which integral I prefer, but I can take no credit for writing it that way. —David Eppstein (talk) 17:47, 2 May 2018 (UTC)

In response to [[ul|David Eppstein], I was well aware that the integral definition was there. However, the definition section should not give a misleadingly incomplete definition. I will restore the rigorous definition. You may make further simplifications as you see fit, but it should not be at the expense of correctness. Alsosaid1987 (talk) 21:27, 2 May 2018 (UTC)

Please see WP:TECHNICAL. It is neither necessary nor a good idea to make our articles harder to read for the target audience (in the case of logarithms, probably secondary school students) by obscuring them with overly-pedantic formalities or detailed listings of caveats, special cases, and exceptions. —David Eppstein (talk) 21:30, 2 May 2018 (UTC)

Non-technical does not mean incorrect or circularly defined. Also, have you actually read the English of the earlier sections of the definition? It is awful and barely comprehensible!

Alsosaid1987 (talk) 21:34, 2 May 2018 (UTC)

Also, tell me one thing about the current version that is pedantic. There is a one sentence alert that the conceptual definition is incomplete, followed by three correct and general definitions.

Alsosaid1987 (talk) 21:36, 2 May 2018 (UTC)

If there is a particular sentence that you take issue with, please point it out. There is no reason why Wikipedia should read like a high school textbook.

Alsosaid1987 (talk) 21:40, 2 May 2018 (UTC)

I would like to point out that more than half the text as it currently stands was pieced together from previous versions. While I instigated the reform of this section, I am by no means hogging the conversation without consideration of previous work. To address your criticism regarding variable name, which seems minor to me, I was following the use of x as the variable as set out in the earlier parts of this definition section. It would be a much more massive change to use t instead.

Alsosaid1987 (talk) 21:46, 2 May 2018 (UTC)

Could you put your signature at the end of the text you've put in or at the very least indent it the same as your contribution rather than having it at the start of a new line thanks. Dmcq (talk) 21:52, 2 May 2018 (UTC)

Maybe you should try diagramming the grammar of the first two sentences of your preferred "conceptual definition" section. That might give you a clue about how likely it is that a student, stumbling over these concepts, would successfully get to the end of one of those sentences. —David Eppstein (talk) 22:17, 2 May 2018 (UTC)

I quote from that version "The logarithm of a positive real number x with respect to base b, a positive real number not equal to 1, is the exponent by which b must be raised to yield x." In case you really are dense, this is the same barely intelligible sentence in your favored reverted version. I changed "by" to "to" to make it at least grammatically correct. As I said, I tried to make minimal changes to respect previous editors. The accidental anonymous edits were mine. Alsosaid1987 (talk) 01:05, 3 May 2018 (UTC)

Brian, I do not understand whether you meant to revert my edits, too, and if so why. Does anyone object to what I added? Dicklyon (talk) 23:29, 2 May 2018 (UTC)

I have not been able to follow the details of the differences between editors here, but I do have the impression that Alsosaid is being disruptive by repeatedly making changes that don't seem to be acceptable to other editors. If I understand correctly, he's trying to make it more rigorous (correct me if I'm wrong). And I agree that more rigor, especially in the early parts of the article, is counter-productive to our purpose of educating people who don't already know this stuff. I did rather like the version that had a "conceptual" section about exponentiation, though, and maybe that was also his? Not sure. Anyway, here's another thing to consider that might lead to yet a different approach: the statement The idea of logarithms is to reverse the operation of exponentiation is not at all historical, if I understand the history right. Rather, the idea of logarithms is to have a function such that log(x*y) = log(x) + log(y); that is, a function that allows one to compute products via addition (using function evaluation via tables). That property leads directly to the log being the inverse of exponentiation, of course, but that may not be a great place to start for people not so familiar with exponentiation. So here's a proposal: let's find some good books that introduce logarithms to high-school students at a pre-calculus level, and see if we find a good approach that we like. Dicklyon (talk) 02:04, 5 May 2018 (UTC)

User:Dicklyon, could you please point out where I've been 'disruptive', and what aspects of the most recent version you've been unhappy with? I've been extremely considerate of previous edits, trying to incorporate as much previous material as possible. However, there were substantial grammatical problems in the definitions section that should be corrected, at the very least. I hope you compare the most recent version and this revert carefully.

Yes, I would like to have more rigor by at least sketching out what the exponential function looks like extended to the reals, instead of leaving it to the reader's imagination. Alsosaid1987 (talk) 02:26, 5 May 2018 (UTC)

As I said, I haven't followed the edits and arguments in detail. What's disruptive is continuing to try to make changes that have been objected to. And why did you now revert my change to the caption, without comment? Is this yet another case of collateral damage based on reverting someone else, or do you actually dislike my addition to the caption? Dicklyon (talk) 02:33, 5 May 2018 (UTC)

User:Dicklyon, do you mean the caption "The special points logb b = 1 are indicated by dotted lines, and logb 1 = 0 is where the curves intersect."? I didn't take this out! This was removed as a consequence of a revert -- I had nothing to do with it (obviously, it wasn't my revert, it was a revert of my edits). Also, I am not the only person to see the value of adding some degree of rigor to the definitions section (please see above, comments by User:Purgy Purgatorio and User:Dmcq), since people other than high school students (college students, grad students) might be using this article as a reference.Alsosaid1987 (talk) 02:45, 5 May 2018 (UTC)

You took out my caption addition here. I'm glad to hear it was not intentional. Dicklyon (talk) 03:14, 5 May 2018 (UTC)

User:Dicklyon, can you explain why you prefer the math template tool to put that in, rather than the TeX editor? For me, the way this displays on my computer is quite unpleasing. If you don't mind, I would like to convert it to TeX. Alsosaid1987 (talk) 03:32, 5 May 2018 (UTC)

This kind of nonsequitur response is also disruptive. You accidentally revert me, and rant about people needing to be more careful about revert. You change math styles and rant about me preferring one over another, when I have not actually changed or expressed an opinion on any such thing (in this caption, I simply copied math expressions as I found them already in the article, without considering what alternatives there might be). Discussion can't converge this way. Dicklyon (talk) 14:49, 5 May 2018 (UTC)

User:Dicklyon, I also find it extremely disrespectful to call someone 'disruptive' based on the opinion of two users, and revert their edits without carefully examining them, thereby wasting someone else's time. The point is to have a conversion about what to include and how to include, rather than calling people out on violating some unstated code of conduct. The revert feature is probably the most powerful tool to be wielded by a wikipedian, and it should be used with care. Alsosaid1987 (talk) 02:53, 5 May 2018 (UTC)

I have not reverted any of your edits, or anyone else's. But if several editors have reverted yours, then you need to work out the issues by discussion, rather than just keeping on editing the article. My impression is that you keep making changes that several editors have objected to (though as I said, I haven't spent the time to be sure of the details, so I could be wrong about what's going on here). Dicklyon (talk) 03:11, 5 May 2018 (UTC)

Please do not base your reverts on "impressions" of my behavior. If you carefully examine what I wrote, you'll see that my subsequent edits were much more conservative and respecting of consensus opinion. However, you made the assumption, as did User:David Eppstein, that my edits were 'disruptive'. Alsosaid1987 (talk) 03:18, 5 May 2018 (UTC)

User:David Eppstein, please explain why there is a need to have equations displayed in two very different formats with the math template. If you can give a valid reason for it, I will gladly concede to the revert. Alsosaid1987 (talk) 02:31, 5 May 2018 (UTC)

I think he reverted more than math formatting edits. It's quite likely that you could do just the formatting edits and get a consensus that that's OK, and then we'd have achieved a positive step. Try that. Dicklyon (talk) 03:11, 5 May 2018 (UTC)

When someone spends the time to make additions to an article, the revert function should only be used when, after careful reading the text, it was judged that the additions are without merit and unacceptable for several good reasons, rather than as a reflex when there's a disagreement with an editor, and you can't point to anything specifically wrong. Alsosaid1987 (talk) 03:23, 5 May 2018 (UTC)

User:Dicklyon and David Eppstein, while there is a need for the article to be understandable by high school students, what is precisely unacceptable about defining what the exponential of a rational number is? After all, these topics are covered in algebra II. It does not hurt to remind people of how the exponential is defined for the rationals. Also, I was following a suggestion by User:Jakob.scholbach to provide a rough definition on the reals by approximating reals with rationals and using continuity. I used concise and elementary language to state this. Alsosaid1987 (talk) 02:38, 5 May 2018 (UTC)

User:Dicklyon, I do agree with your point that the central defining identity for logs should be log(xy)=log x+log y, as that is the whole point of having a log table. Alsosaid1987 (talk) 02:58, 5 May 2018 (UTC)

I am not doctrinaire or a zealot, as some of you have portrayed my edits as being -- User:Jakob.scholbach has long ago convinced me that it was a bad idea to give the full "elementary" construction of a real exponential function. However, I still think the integral definition of the log should be given more prominence, since, for all intents and purposes, it is concise and rigorous, given a definition of the Riemann integral. I would be willing to debate this point. Alsosaid1987 (talk) 03:11, 5 May 2018 (UTC)

If it's just changing formatting from one format to another (html to math or vice versa) I am also likely to revert it. We don't need repeated back-and-forth on that point until such time (not now) as there is a clear choice of which format is better. —David Eppstein (talk) 04:02, 5 May 2018 (UTC)

So you are okay with the ugly, incongruous display, containing a hodge-podge of TeX and Math Template, despite the fact that TeX is clearly much more expressive and a lingua franca for scientific publishing? Alsosaid1987 (talk) 04:11, 5 May 2018 (UTC)

No, if the formatting is inconsistent I'm ok with changing it to become more consistent. And if there's a reason why one formatting won't work at all (as the math templates don't work in italic reference titles) I'll change them myself. It's the changing of consistent formatting from TeX to html or vice versa for no reason than a preference for one over the other (I happen to think that the Wikimedia software's rendering of TeX is also ugly) that I'm likely to revert. —David Eppstein (talk) 04:17, 5 May 2018 (UTC)

User:David Eppstein, in case you haven't noticed, there is no rhyme or reason as to why some mathematical expressions in this article are in math template or in TeX, except maybe the math template ones are simpler, because it is a crippled system. So, no, it is not consistent. I haven't heard you make a single valid argument contra my edits. It's apparent that you have a desire to not let me alter the article because you seem to see me as a troublemaker. Just who made you the policeman of wikipedia? I refrain from using uglier language out of respect.Alsosaid1987 (talk) 04:32, 5 May 2018 (UTC)

I think that it's typical in articles to have displayed equations use TeX, and inline expressions and equations to use HTML. Whether this is really best, and whether it's done consistently in this article, I'm not sure, but to say there's no rhyme or reason may not be quite right. Dicklyon (talk) 14:52, 5 May 2018 (UTC)

I agree with Dicklyon that it is typical to use TeX/<math> for displayed equations and {{math}} for inline text, and at this point, I'd prefer to see inline text stay as {{math}}. Initially, WP turned TeX markup into a bitmap image; its baseline alignment was poor, its size did not match the text, and copy-paste operations could not get any text out of the bitmap. The {{math}} template does a better job at that. There have been improvements to TeX markup display (see user preferences), but browser/MathML support is an issue. Glrx (talk) 18:00, 6 May 2018 (UTC)

I apologize in general for anything untoward I said in these arguments. What do I know? I'm just an organic chemist. However, I care deeply about the quality of Wikipedia. I can safely say that I've learned more from Wikipedia than any other resource, and spend about 10-15% of my waking moments on it since my days as an undergrad. Jimmy wales's invention was literally life-changing.

For an article of this importance, I was more than a little concerned by the quality of the prose -- in particular, the the rambling and illogical order of sentences of the definition section, in addition to my perception that it's written in a way that caters to a superficial and high-school level understanding. The formatting of the mathematical expressions is a secondary concern. I will leave this article alone, but I hope others pay attention to its quality and care enough to not be complacent with whatever current state it's in. Best wishes, ymw Alsosaid1987 (talk) 04:11, 5 May 2018 (UTC)

The complex details

First of all, I think "i, the square root of -1" should be deprecated and eradicated now and for all times in the future. It should be for math reason "i, that squares to -1", and, in emergency cases, at least "i, a square root of -1".

The article referred in several places to "Log" with capital L. This was an intentional, if not also addressed, habit to refer to the function, yielding a somehow defined principal value of the complex log (whatever one likes to call this). I think this should not be swept over board, together with other changes, which do not survive in consensus, dominated by some.

Just another, somehow complex detail, but not within the complex numbers, is the typesetting of the name of Euler's constant. Should it be in "roman" or in "italic"? Purgy (talk) 07:59, 6 May 2018 (UTC)

Thanks, I have fixed these. Jakob.scholbach (talk) 12:35, 6 May 2018 (UTC)

Thanks, I totally missed the point of the capped Log; thanks for making it explicit in the text now. As for your italicizing e, it appears many times roman still, which I thought was what we had decided (or at least someone had pointed out) above. I think I had it consistently roman, but let's decide before going back there. In my own book, I use italic. Dicklyon (talk) 16:37, 6 May 2018 (UTC)

It looks like we had e consistently italic in the past; I'll fix it back to that. Dicklyon (talk) 17:46, 6 May 2018 (UTC)

Italic is how it almost always appears in professional mathematics. —David Eppstein (talk) 17:55, 6 May 2018 (UTC)

And I'm sorry I misinterpeted something and messed that up. All fixed now, I think. Dicklyon (talk) 18:18, 6 May 2018 (UTC)

In the complex, I've done some cleanups to use φ inline, rather than φ or φ, but it still doesn't match TeX math $\varphi$ (\varphi), at least not in my font. Should we use $\phi$ (\phi) instead? Better ideas? Dicklyon (talk) 17:01, 6 May 2018 (UTC) I've also filled in some missing steps in the complex section, hoping to get to where someone who doesn't already know this stuff might be able to understand it. Dicklyon (talk) 17:41, 6 May 2018 (UTC)

Sorry, my personal preference is using <math>...\varphi...</math>, because any sensible formula requires LaTeX, but this has no broad consent, and has been eradicated. As long as there is no consent about how the log is to be defined for this article, I do not think much about the lines along which the complex section should look like. To be honest, I saw rather a degradation for the moment.

The sections about Motivation and Definition are, in my view, noticeably more degraded than improved. Taking "10" as an additional paradigm enforces the unlucky mixup of numbers and their ubiquitous decimal representation, instead of making any math perspective more accessible. Why should anyone believe that raising 10 to some power has more interesting properties than any other number? I pity the harsh reversion of Alsosaid's efforts more than I would advocate the current state.

Just my opinion, I'll try to keep clear of any edit controversy. Purgy (talk) 13:41, 7 May 2018 (UTC)

Opinions are OK; let's have more and discuss. And thanks for your tweaks to the complex section. Re the 10, I think there's a very good reason that that's "common" and that students are introduced first to base 10 logs, for their not-coincidental relationship to how we write numbers in decimal. Starting abstractly, as opposed to with a special base, may work for a mathematician, but not for the typical reader who wants to know what a logarithm is. Dicklyon (talk) 14:16, 8 May 2018 (UTC)

Thanks for the nice invitation to discussing and for welcoming my complex tweaks.

Well, ... I do not think that there is a "very good reason" for a Motivation via 10 anymore. To the contrary, I insist on it having detrimental effects, mentioned above and acknowledged by jakob.scholbach to at least marginal extent. The reasons to employ 10-logs are bound to the historic common use of decimals in any number crunching by hand (BTW, its "by foot" in German) and the associated, manageable handling of the required normalisation, when applying tables. Both conditions have mostly vanished: thanks to electronic calcs tables and number crunching are totally(?) out of focus in math education. Additionally, I think that here the typical reader had enough of 10s after the gentle lead, and, with targeting an encyclopedic treatment of the logarithm, a certain positive slope in the learning curve is necessary.

Alsosaid was not the first editor (e.g.: Nageh) of this FA to hint to a more stringent "definition" of the log, but his efforts were eradicated (without discussion!), even though Jakob.scholbach -an opponent of Nageh's, mine, and Alsosaid's proposals- had tried to initiate a discussion, and Alsosaid's most perplexing positions had been moved down and partly removed, and he himself had withdrawn his first attempts in part. I won't comment on the following phase of retaliation and brusqueness, besides me still having reservations to partake in a discussion here.

To me, my primary open topic, about how to "formally introduce" (I avoid "define") the logarithm in this article, or mention the existence of different possibilities to do so, is still unresolved. I regard the "inverse of exponentiation" as the silver bullet for this article, but I dislike strongly the intransparent use of the real numbers. This may be appropriate/unavoidable(?) in High School math education, but a serious encyclopedia should not blindly follow this dulling path, it must -at least- mention the subtleties, IMHO. Purgy (talk) 09:02, 9 May 2018 (UTC)

The definition

Does anyone really think

The logarithm of a positive real number x with respect to base b, a positive real number not equal to 1, is the exponent by which b must be raised to yield x. In other words, the logarithm of x to base b is the solution y to the equation

$b^{y}=x$

The logarithm is denoted "log_b x" (pronounced as "the logarithm of x to base b" or "the base-b logarithm of x" or (most commonly) "the log, base b, of x"), such that the defining identity from above becomes

$b^{\log _{b}x}=x$

In the equation y = log_b x, the value y is the answer to the question "To what power must b be raised, in order to yield x?".

is a clear way to define the logarithm? I happen to find this rambling definition to be repetitive without being clear. Grammatically, things are raised "to" an exponent, not "by", as the first sentence implies. Also b^y=x is not an identity, as the second sentence implies. And what does the third sentence really add? I would prefer the addition of a rigorous definition somewhere, and I think some of you also support that. I won't make any changes because I've apparently become a persona non grata when it comes to editing this article. I think User:David Eppstein should in particular be embarrassed for refusing to consider any changes to this status quo. Alsosaid1987 (talk) 06:58, 5 May 2018 (UTC)

I agree that this is a poor way to define logarithms, and that we should discuss a way to do better. Discussion is more likely to converge that long sequences of edits and reverts from which editors are supposed to infer ideas and intent. Dicklyon (talk) 14:56, 5 May 2018 (UTC)

My feeling is that the prose could be improved but that, among choices of definition (e.g. versus the integral formula or the limit of exponents of fractional powers) this one is the simplest and best. It depends on already having in mind a definition for exponentiation, but that's ok. —David Eppstein (talk) 15:58, 5 May 2018 (UTC)

As long as there is recognition that something can be improved, and that this cannot be the final form of the definition section, then I've done what I wanted to do. I really don't care about this page per se, but the behavior of certain editors and their desire to silence 'dissent' because this is an FA (I cannot understand how the current form of the article would merit it) reveals that most of the criticisms about the current editorial system have been on target. I guess that makes me sad. Alsosaid1987 (talk) 20:40, 5 May 2018 (UTC)

The prose could be improved (perhaps "the defining identity from above" should just be "the previous equation", for example), but in my view, this definition is the correct one to use at this point in the article. XOR'easter (talk) 20:56, 5 May 2018 (UTC)

A final point that may have escaped the attention of others -- does anyone actually have access to or even know what the treatise cited as ref. 1 is? I don't find it to be notable or accessible enough to be included as a reference. Giving a single example of the exponential function at a non-natural number value (and not at a general rational number but at -1) does not, in my opinion, give people any clue how they are defined on the reals. I still find the definition woefully incomplete. 67.186.58.77 (talk) 21:11, 5 May 2018 (UTC). Was signed out. Alsosaid1987 (talk) 21:12, 5 May 2018 (UTC)

Google books shows snippets of where ref 1 talks about extending exponentiation to non-integers, but I don't have and it costs over $100 to get a used copy, so I'll just trust it; there's no real need for it to be notable or more accessible. The way it's referenced is lame though, so maybe I'll fix that. For an example, something like an exponent of 1/2, corresponding to a square root, might be more helpful and relatable. I expect you could work on such things productively if you want to. But go slow, because if you rack up a pile of changes that on balance don't look great, some editors are more likely to revert you than to try to sort out exactly which bits are OK and not. Dicklyon (talk) 21:58, 5 May 2018 (UTC)

This is the wrong article for going into details about the definition of the exponential function, but my preferred definition for that would be the power series. The definition can then be extended to other bases by the change of base formula with the observation that the result agrees with the naive definition for integer powers. I think that's better than trying to use limits to define exponentiation for arbitrary bases without regard to the base. It does mean that you have to define the natural log first (so you can do the change of base) before you get to exponentiating with other bases and taking logs by other bases, but I don't really see the need to go into all that in an article introducing these basic concepts. —David Eppstein (talk) 22:06, 5 May 2018 (UTC)

Dicklyon, thank you for the civil response and suggestions. I again apologize for any unwarranted accusations I made against you. With a certain degree of cynicism, I find that it is much easier to play armchair Wikipedian and point out flaws rather than to correct them -- at least for now. Though a lifelong devotee of mathematics, it's hardly my day job, and there are areas of Wikipedia closer to my area of expertise to which I could more actively contribute. Cheers, 67.186.58.77 (talk) 22:25, 5 May 2018 (UTC) Ah, logged out again. Alsosaid1987 (talk) 22:26, 5 May 2018 (UTC)

By the way, the motivation and definition section is nearly unchanged from when User:Jakob.scholbach added it in 2011 in the this edit. I agree it's still awkward and could use a rewrite. Why don't we intertain proposed rewrites here and see if there's one we prefer? Dicklyon (talk) 22:11, 5 May 2018 (UTC)

Another somewhat nutty bit there is the "It follows" that came in with this 2011 edit by User:Randomblue. It looks to me like there hasn't been enough of a definition by that point to say what follows. It would be better as it was before, more like a motivating definition that a deduction: "The logarithm of 8 with respect to base 2 is 3, since 2 raised to the third power equals 8." Dicklyon (talk) 22:37, 5 May 2018 (UTC)

I've added some motivating/historical/alternative definition stuff there, as a trial balloon. I won't be offended if someone wants to revert it, but I hope they'll consider and maybe make a different try if this one is not in a good direction. Dicklyon (talk) 05:51, 6 May 2018 (UTC)

Been there, done the reversion. I think in times of pocket calcs the tedious arithmetic of former times is no good motivation for nowadays students. Furthermore, explicit voices were raised above, for not using historical access to problems, when more modern, and since simpler ways are accessible, but rather report them in History sections. Purgy (talk) 07:42, 6 May 2018 (UTC)

OK, no history in the motivation then. I put in some more minor rewordings; see what you think. Dicklyon (talk) 16:34, 6 May 2018 (UTC)

Instead of jumping into a general definition, maybe start with any positive number x can be represented as 10^y. The common logarithm of x is the exponent y: log(x)=y. Then generalize to nonzero base b: log_b(x). Then the natural logarithm uses base e. If the historical motivation is out, then offer some motivation to use logarithms: logarithmic plots map values so harmonics are spaced at equal intervals rather than ever increasing intervals. Glrx (talk) 18:21, 6 May 2018 (UTC)

Let me know if you like what I did with that powers of 10 idea in the article; or tweak it to be more like you had in mind. I don't see what you have in mind about harmonics, frequencies of which are in arithmetic progression already without a log. Dicklyon (talk) 00:43, 7 May 2018 (UTC)

Yes, that is the idea.

Brain bubble; I meant octaves. I was thinking of the illustration at Logarithm#Music.

Glrx (talk) 02:28, 7 May 2018 (UTC)

I am not sure the current content is so good: it says motivation and definition, yet gives little or no motivation. It uses the notation log x for base-10-logarithm, yet in the remainder of the article we usually explicitly point out the base so as to avoid hurting predilections of some branch (math, phys, computer science...). It starts out with a non-trivial fact (10^x=y can be solved for any y), before even (roughly) explaining what 10^x means.

"Logarithms do not require integer exponents; they exist for any positive number. " -- what exactly does "positive number" refer to?

What I do like about the approach is to highlight (using a low-key language) that logs are an inverse operation. We could do this: the operation "adding 10" is undone by "subtracting 10". The next step in arithmetic is "multiplying by 10" (or maybe 3, Purgy Purgatoriois right in pointing out that using 10 has questionable consequences, we don't want to highlight the number 10 here, but really any number). It is undone by "divide by 10". Finally, there is the operation "raise 10 to some power". Logarithms (to base 10) undo this operation. Jakob.scholbach (talk) 21:10, 7 May 2018 (UTC)

I have implemented a motivating approach to logs as an inverse operation. I have also (re?)added a short explanation on b^y for arbitrary y. Jakob.scholbach (talk) 19:38, 9 May 2018 (UTC)

I think I get your intentions, but cannot feel motivated in honesty by comparing "inverse operations", what ever this is beyond elementary math conceptions, with –in the accepted nomenclature– the "inverse functions" of a family (parametrized by the base) of most well behaved functions. Your motivational effort is to me in no way better than the, meanwhile often mentioned, intransparent application of the Main Theorem of High School Math. Ignore to your likings. :) Purgy (talk) 07:13, 10 May 2018 (UTC)

I don't know what the "Main Theorem of High School Math" refers to, but I rather like what Jakob has done. And I agree, too, that Alsosaid had some good points, but got himself pushed out by being too pushy himself. If you want to make a more specific proposal for an introduction or definition section, you can do that here, or on a user subpage, or in the article itself, to get a reaction from other editors; but if in the article itself, the probability of revert remains high. Personally, I have a hard time imagining coming to an understanding of logarithms, or even exponentiation, without starting with powers of 10. Certainly it's possible and logical, but without the help of visualizing by how we write numbers, it's too abstract for most naive people to make the leap. Dicklyon (talk) 14:27, 10 May 2018 (UTC)

Napierian logarithm

In the article Napierian logarithm, I think help is needed. The development there started with this anon edit that I cannot find any support for in sources. One can easily construct a story in which it's true (and I have coded up such a construction), but when I read detailed accounts in sources, it seems to be not quite right. I haven't read the Constructio myself yet, which I might have to resort to to try to interpret the difference in the points of view, the alternative being that the Napierian log of x is simply $-10^{7}\ln(x/10^{7})$ (a very subtle difference but affecting the quoted numbers a little). See the ref I added there recently. Dicklyon (talk) 14:29, 8 May 2018 (UTC)

See Talk:Napierian logarithm. Dicklyon (talk) 14:28, 10 May 2018 (UTC)

Cite tag

@Purgy Purgatorio: Greetings! I'm not sure what you meant in the edit summary of this revert: "we had this already"...what is "this" referring to? As far as I can tell from reading Wikipedia:HTML 5#cite, 1. the <cite> tag should be applied to the text that describes the cited work, and 2. it shouldn't be used in articles and we have templates to use in its place. This article has markup like this:

<cite id="labegarithmProducts"><math>\log_b(x y) = \log_b x + \log_b y</math></cite>

which renders as:

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

This markup appears to be broken, as it isn't displaying anything to the reader that indicates which source supports this claim. Am I missing something? Thanks! -- Beland (talk) 19:59, 27 February 2019 (UTC)

Beland, "we had this already" refers to the repeated problem of this tag not being useful in locating the problems, a fact "we had already" explained. I have repaired the addressed HTML flaw. Purgy (talk) 20:24, 27 February 2019 (UTC)

@Purgy Purgatorio: Ah, yes, some other users on other pages have also been asking how to find the tags that are causing the problem. That's why I added an HTML comment and edit summary about <cite>, but perhaps it was insufficiently clear. This page actually still has instances of <cite> that need to be cleaned up. Wouldn't we prefer to convert them into <ref> tags instead of removing them completely? -- Beland (talk) 20:38, 27 February 2019 (UTC)

@Purgy Purgatorio: Vector space also has <cite> tags that should be converted (I assume) into visible footnotes, so I don't understand removing the cleanup tag? -- Beland (talk) 06:30, 28 February 2019 (UTC)

Sorry, I'm too slow. Plastering otherwise fine articles with this tag is a heavily disproportionate deprecation of these articles. My suggestions are either to repair HTML-flaws yourself, when they pop up at a systematic search, or -with strong reservations- by a bot, or create some less devaluating hinting mechanism (category of articles with HTML-flaws?) and add this to a far less prominent place within the article. Purgy (talk) 07:34, 28 February 2019 (UTC)

@Beland and Purgy Purgatorio: disambiguators and people fixing lint errors had good luck posting a list of affected articles to WT:WPM. (I agree that the tag seems a very ugly way to address a minor issue.)—JBL (talk) 11:01, 28 February 2019 (UTC)

Unfortunately, a bot cannot reliably fix all the cite tag problems correctly (at least not without investing even more programming time than it would take to fix the problems manually) since the way they are being misused varies somewhat between articles. I ran a script that identified over 10,000 articles with some sort of undesirable HTML markup, so it's not feasible for any one person to fix them on their own. It sounds like you don't want this tag to exist at all, though it's been used by Wikipedia:WikiProject Wikify since 2010, and there are lots of similar tags that put articles somewhere under Category:Articles covered by WikiProject Wikify for relatively minor or major issues. I'm certainly sympathetic to the argument that this looks a bit alarming on an otherwise tidy article, though on the other hand, it also significantly raises the chances that an editor with an interest in the article will fix the problem. (It's also used on a lot of untidy articles.) I'm open to a change in the process if the WikiProject wants to do that. In the meantime, I'll at least try to improve the instructions on the template so editors can more quickly and easily address identified problems.

For this article, I just removed the <cite> tags. It looks like they were actually being used as HTML anchors and not for citations as the HTML standard intends, but there isn't anywhere in the article that's actually linking to the anchors anymore. -- Beland (talk) 18:02, 28 February 2019 (UTC)

'*giggle* I had my reason why I rm just one cite. :) One must really dig into an article to do this clean up. May I remark: a specification of the flaws and a count to repair might be attractive. How about creating subcategories to put the respective articles in? They are by far less annoying than that tag. Purgy (talk) 18:26, 28 February 2019 (UTC)

Logarithms of negative numbers

The graph of the Logarithm of a number correctly shows that you cannot express the log of a negative number in terms of a real number, however it appears to me that this can be circumvented if we resort to complex numbers.

Since e^iπ = -1

 ln(e^iπ) = ln(-1)
 iπ = ln(-1)

But 3e^iπ = -3 So ln(3e^iπ) = ln3 + ln(e^92.25.126.173 (talk) 12:06, 23 November 2019 (UTC)iπ) = ln(-3)

 92.25.126.173 (talk) 12:07, 23 November 2019 (UTC)ln3 +  iπ = ln(-3)

which implies that in general ln(-R) = ln(+R) + iπ

We therefore have a consistent way of representing the log of negative numbers in terms of complex numbers (the real number ‘R’, the imaginary number ‘i’ and the transcendental number ‘π’). Graphically the log of a negative number is equal to the reflection of its positive log which is then ‘lifted’ in a vertical imaginary axis to the number iπ. That is the logarithmic graph of a negative number mirrors that of its positive version but is lifted in the third dimension into the imaginary plane by a constant value of π. However note that 3iπ = 3ln(-1) = ln(-13) = ln(-1) that is ln(-1 ) can also equal 3iπ or more generally R(iπ) =ln(-1) In other words ln(-1) can be equated to a series of integers on the complex (vertical axis). Hence this function of the logarithm of a negative number is multi-valued but is discontinuous at zero. [We could perhaps name this function ‘nelog’ for negative logarithm] I am not sure if this has been written about before, if so perhaps someone could supply further details. Paul 92.25.126.173 (talk) 12:05, 23 November 2019 (UTC)

But Wikipedia does not allow wp:original research. Also, on this talk page we can only discuss the article, not the subject—see wp:Talk page guidelines. If you have a proper book reference (wp:reliable source) that covers this, we can of course discuss here whether we can put something from it in the article. - DVdm (talk) 12:10, 23 November 2019 (UTC)

Also, this is discussed and generalized in section "Complex logarithm". D.Lazard (talk) 12:41, 23 November 2019 (UTC)

Alternative notations

Should this article mention alternative notations for logarithms? Such as the "triangle of power" described here or the upside-down radical symbol. I feel like they might be worthy of at least a brief mention in the article. (I am not suggesting that they be used throughout the article) --NeatNit (talk) 23:51, 23 February 2020 (UTC)

Covid-19 R0 and Re example?

1. Re = effective reproduction number, sometimes also called Rt, is the number of people in a population who can be infected by an individual at any specific time. It changes as the population becomes increasingly immunized, either by individual immunity following infection or by vaccination, and also as people die; and
2. R0 = the basic reproduction number is defined as the number of cases that are expected to occur on average in a homogeneous population as a result of infection by a single individual, when the population is susceptible at the start of an epidemic, before widespread immunity starts to develop and before any attempt has been made at immunization - so if one person develops the infection and passes it on to two others, the R0 is 2.

Can anybody help how these Ro and Re fit in a logarithm formula? Sincerely, SvenAERTS (talk) 09:08, 19 July 2020 (UTC)

Formulas are more commonly given as exponetial than as logarithmic, using exp(R * t) for some concept of time t. These parameters don't define a time scale, though, so it's hard to actually put them into a function of time. You can find exp(a*t) where a depends on Re or R0, and a > 0 when R > 0. Plotting the logarithm of that function makes a linear a*t curve if a is constant. Dicklyon (talk) 02:26, 23 July 2020 (UTC)

Need a section on units?

There seems to be a fair amount of controversy around the web on if you can take the logarithm of a number that has units. A web search with query like "units in logarithms" returns many pages that all seem to be wrong. In general, the claim is basically that you cannot take the logarithm of a number with units. Period. For example at .

References

This seems incorrect to me. If the Log is considered the integral of dx/x then the units of the integral are the units of that quantity. An integral is a sum with the units of the individual terms. In the case of dx/x the result is always unitless since dx and x have the same units so the ratio is always unitless. So Log10(100meters) is 2 and no rules are broken. That also means though that the Log is a lossy transform since you cannot recover the unit with exponentiation. E.g., 10^(Log10(100meters))=100 and not 100meters.

There are many examples in chemistry, physics and engineering where the logarithm is taken of a quantity with units, for example the Arrhenius Equation: deltaG = RTln(k) where k may have units.

Some claim that there is an implied de-dimentionalization when taking a Log. Using the above example, people claim that what is actually being done is Log10(100m/1m). This seems to be incorrect and unneeded. Calculus works just fine with units and the Log function is no exception. It is lossy, but many mathematical transforms are lossy (e.g., sqrt(x^2) != x for all x).

So perhaps a section can be added to this article that discusses units in logarithms.

Jsluka (talk) 18:47, 10 July 2020 (UTC)

You can meaningfully take a difference of two logarithms of quantities with the same units, just as you can meaningfully take a ratio of quantities with the same units and get a dimensionless result. —David Eppstein (talk) 20:30, 10 July 2020 (UTC)

Jsluka (talk) 21:58, 22 July 2020 (UTC) Exactly, so by definition you can take the logarithm of a quantity that includes units and you (1) aren't breaking any rules and (2) get a unitless result.

NO. You can meaningfully take a DIFFERENCE of logarithms of two quantities that are measured in the same units. The logarithms themselves are only defined up to an additive constant (just like indefinite integrals are defined only up to an additive constant). —David Eppstein (talk) 22:22, 22 July 2020 (UTC)

Agree, sort of. You can take the log of numbers that have units, and those units affect the result, which is nevertheless unitless, the ratio of the number to the unit (that is, think of the argument as a unitless ratio, with the unit as the reference). The differences of such things (where the original numbers have the same units) are indifferent to the units. In any case, the logs themselves are unitless, unless you adopt the position that natural log has unit 1 and other logs have different units, which is also a valid viewpoint. Dicklyon (talk) 02:20, 23 July 2020 (UTC)

If you say that a certain distance is 10, or that a certain volume is 100, the result is meaningless unless you multiply it by its unit. If you say that a (decimal) log of a certain distance is 1, or that the log of a certain volume is 2, then it's equally meaningless unless you add it to the log of its unit. So you could reasonably say that log₁₀ (100 liters) is 2 + log(liter). When you subtract the logs of units cancel, just like when you divide numbers with the same multiplicative units the units cancel. —David Eppstein (talk) 06:07, 23 July 2020 (UTC)

A prime example. maybe worth including, is dBm, the absolute power level referenced to one milliwatt. The difference between 30 dBm and 40 dBm is 10 dB, a dimensionless ratio. —agr (talk) 07:40, 23 July 2020 (UTC)

I figured out the idea of additive log units in high school, but was later convinced that we are not supposed to use them. Equations are supposed to be arranged such that only the log of appropriately unitless combination is used. Arrhenius is an interesting example. While in theory k has units, in practice it is considered arbitrary. In theory the slope depends on an activation energy, but the only measurement of that energy is the Arrhenius plot itself. For log graph paper, the graph value is the log of the desired quantity divided by the value on the graph axis tic mark. One that I always found interesting, is radioactivity uses base 2 log, unlike just about everything else. For most things, actual measurable quantities go into the exponential. A capacitor discharges as exp(-t/RC) where we can measure R and C. But for radioactive decay, there is unmeasurable physics inside the nucleus, such that we only measure the decay time. But it is really easy here. You need to find a WP:RS if you want to add it. Gah4 (talk) 09:00, 23 July 2020 (UTC)

easy

The article says that logarithms are less easy than root. Considering that the common way to compute roots, especially non-integer roots, is with logs, that isn't so obvious. Should we be judging what is easy and what isn't? Gah4 (talk) 22:13, 10 October 2020 (UTC)

I cannot find in the article any assertion comparing the difficulty of computing logarithms and roots. What I see in the lead is the assertion that before the invention of computers, logarithm tables made scientific computation easier than before. This is blatantly true. Nevertheless the article contains too many occurrences of "easy" or "easily", per as MOS:EDITORIAL. D.Lazard (talk) 09:45, 11 October 2020 (UTC)

Original research?

Dear Wikipedians, as part of the development of the Vietnamese version, I've read this article carefully and now I'm afraid that there are some original research, as follows:

1. There are also some other integral representations of the logarithm that are useful in some situations ... with two integral representations of ln(x) --> not sourced
2. The Taylor series of ln(z) provides a particularly useful approximation ... less than 5% off the correct value 0.0953. --> not sourced
3. A closely related method can be used to compute the logarithm of integers ... to the end of the section --> not sourced
4. The non-negative reals not only have a multiplication, but also have addition, and form a semiring, called the probability semiring; this is in fact a semifield. The logarithm then takes multiplication to addition (log multiplication), and takes addition to log addition (LogSumExp), giving an isomorphism of semirings between the probability semiring and the log semiring. --> not sourced

These were not presented in the 2011 featured version. Another problem: at least for me, the "Motivation and definition" read more like a textbook rather than an encyclopedic article. Are them serious issues? I'm tagging Jakob.scholbach, the nominator for FA back in 2011, and also the WikiProject Mathematics for rapid response. (Note that I'm not a contributor of this article.) Thuyhung2112 (talk) 15:25, 10 October 2020 (UTC)

This may or may not help your question, but in many scientific articles WP:CALC applies more than WP:OR. You might also read WP:SYNTHNOT. For example, many mathematical operations are so common that many textbooks will have them, and so no need to site an actual reference. All this is not necessarily agreeing or disagreeing with what you say, but to keep discussion going. Gah4 (talk) 22:07, 10 October 2020 (UTC)

I think calling these citations original research is a bit exaggerated. Some of these sentences could use an additional reference (or in some cases I would maybe merge them somehow differently in the surrounding text), but OR is a bit of a longshot here. Back from 2011, when I nominated this article, it has in a few places mildly deteriorated (IMO), and some of the spots you mention belong to the additions that I personally consider not always helpful. But in any case this is not OR proper, IMO. Jakob.scholbach (talk) 12:30, 11 October 2020 (UTC)

Some of this was also discussed at the ref desk. This is starting to run afoul of WP:TALKFORK, but to summarize, I'm inclined to remove the integrals: they're just two specific cases of a more general result, any of which involves a logarithm. There are lots of other integrals where logarithms pop out unexpectedly. The claim that these are "useful" is dubious, and this much at least does need to be clarified and sourced. It's also poorly explained, but unless has any insights into this, I think it would be better to just axe it. –Deacon Vorbis (carbon • videos) 13:26, 11 October 2020 (UTC)