Wikipedia:Reference desk/Archives/Mathematics/2015 February 19

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February 19

Logarithms

What Exactly Are Logarithms And How Do You Solve For Them. They Really Confuse Me As They Would To Any One Else But This Defines It.

[Redacted copyright violation: 1st 6 paragraphs from S.O.S. Math logarithm page. -- ToE 13:20, 19 February 2015 (UTC)]

— Preceding unsigned comment added by Yezenia Lopez (talk • contribs) 00:02, 19 February 2015 (UTC)

I am confused. What is the question again? 175.45.116.60 (talk) 04:08, 19 February 2015 (UTC)

This person appears to have just posted a few paragraphs from a textbook, except that some mathematical expressions have been replaced by some sort of internal code ("tex2html_wrap_inline11" obviously means 3×5 or perhaps 5×3). --70.49.169.244 (talk) 11:25, 19 February 2015 (UTC)

Oh, wait, I get it now. The first boldface line is the question. What are logairithms and how do you solve for them? The rest is what the person has in their textbook and they aren't being helped by it. --70.49.169.244 (talk) 11:27, 19 February 2015 (UTC)

One place to start within Wikipedia is by looking at our page on logarithms. In particular, read the definition. You'll see that "logarithm" (usually we just say "log") is just a shorthand way of talking about a relationship between numbers. We know that

$2^{3}=2\times 2\times 2=8.\,$

so if someone asked you "what is $x$ in this equation

$2^{x}=8\,$  ?"

you could immediately answer "3".

When someone asks "what is the logarithm to the base $b$ of $y$?" they are simply asking "what is the $x$ in this equation:

$b^{x}=y\,$  ?"

We can write this as $\log _{b}y=x$, which we read out loud as "log to the base b of y equals x", so "log to the base 2 of 8 equals 3".

A good thing to do is to practise this with whole lot of examples, like this:

$3^{2}=9$,  so  $\log _{3}9=2$   "log to the base 3 of 9 equals 2"

$3^{3}=27$,  so  $\log _{3}27=3$   "log to the base 3 of 27 equals 3"

$3^{5}=243$,  so  $\log _{3}243=5$

$5^{2}=25$,  so  $\log _{5}25=2$

$5^{3}=125$,  so  $\log _{5}125=3$

$5^{4}=625$,  so  $\log _{5}625=4$

$10^{1}=10$,  so  $\log _{10}10=1$

$10^{2}=100$,  so  $\log _{10}100=2$

$10^{3}=1000$,  so  $\log _{10}1000=3$

and so on...

Now we come to the clever bit: we know that $9\times 27=243$. Look at the list above and watch what happens with the logs... $\log _{3}9=2$ and $\log _{3}27=3$, and $\log _{3}243=5$. Notice that $2+3=5$. It turns out that this always works (so long as the logs are to the same base), so when we multiply two numbers together, it's "the same" (for mathematicians) as adding their logarithms. This is why logarithms are so important: they give us a way of turning questions about multiplication into questions about addition, and most of the time it's easier to do addition than multiplication, so it's a way of making everyone's life easier. In fact, in the days before calculators, people would regularly use logarithms to do multiplication. You would look up the logs of the two numbers you wanted to multiply in a pre-printed book (called a "Table of Logarithms"), add the logs together, then go back to the book and find which number's log was the result of your addition, and this would be the result of your multiplication. This sounds cumbersome to us, but right up to the 1970s this is how engineers and scientists all over the world did their multiplication, and had done for centuries.

There's quite a lot more to say, but I hope this at least gets you started in the right direction.

You should also take a look at this set of videos: [1]. If you're studying at school, it's also usually a good idea to talk directly to your teacher (before or after class works well, in my experience!), and explain that some things are unclear to you. The teacher's job is to help you understand, and most teachers would rather their students ask questions, because if they haven't been clear enough to help you, they probably haven't been clear enough to help the other students either. If you're studying alone, a good idea is to search for videos on YouTube (like the Khan Academy ones above), and watch as many as you can, because each teacher has a different way of explaining things, and eventually you'll pick it up. Hope this has been of some help. RomanSpa (talk) 13:42, 19 February 2015 (UTC)