Math stuff and other things...but mostly math.: Logarithms

So here's some stuff about logarithms. I assume you know all of the basic properties of logs. If you don't, go look them up here:

http://www.artofproblemsolving.com/Wiki/index.php/Logarithms

Or anywhere else you see fit.

The following type of problem is incredibly common:

If $\log_{8}3 = P$ and $\log_{3}5 = Q$ , express $\log_{10}5$ in terms of P and Q

In log problems where the bases are different, we usually have to use change of base formula. Remember, the base we are changing to doesn't HAVE to be the common base 10 (that's the way it is usually taught in school, so logs can be estimated on the calculator). It can be any base, as long as the bases are the same in the numerator and denominator of the changed expression.

Seeing that the expression for Q and the expression we are trying to simplify are both taking the log of 5, we should change to the base of Q, which is 3.

$\log_{10}5 = \frac{\log_{3}5}{\log_{3}10} = \frac{Q}{\log_{3}10}$

Notice that 10 = 5 * 2. We can rewrite the bottom logarithm to include another Q:

$\frac{Q}{\log_{3}10} = \frac{Q}{\log_{3}2 + \log_{3}5 } = \frac{Q}{\log_{3}2 + Q }$

Noticing that the log contains a 3, changing to base 8 will allow us to include a P. Also, 2 is a power of 8, so we will be able to eliminate the log completely.

$\frac{Q}{\log_{3}2 + Q } = \frac{Q}{\frac{\log_{8}2}{\log_{8}3} + Q } = \frac{Q}{\frac{1/3}{P} + Q } \Rightarrow \frac{Q}{\frac{1}{3P} + Q }$

I know that last expression as the solution is a bit ugly, but I don't think it simplifies.

A lot of times, with those kind of log problems, you really do have to play around with it, or "guess and check" (I hate, hate, HATE guess and check). Obviously it's not all guess and check. Being able to recognize when certain properties can be used to simplify it is key. But usually you're not really heading for the answer right away, if you know what I mean. Rather, you're just sort of tinkering with it until it clicks, and then you start heading for it. I'm sure this method applies to other problems too, but I find it noticeably with these sort of "this is p, this is q, whats this in terms of p and q" problems.