Better to memorize logarithm rules? And how? Do people good at math totally memorize these logarithm rules below? If so, are there good mnemonics for this? I'm bad at math and I only memorize these rules really vaguely by rote, thus when needed, I'm not sure if I remembered them correctly. So I always end up checking if they are true by testing them assigning numbers like 2 to m and n on the fly. Is this something smarter people would think is a stupidly inefficient process to go through? I wonder how other people deal with this?


$\log_b (mn) = \log_b(m) + \log_b(n)$
$\log_b(m/n) = \log_b(m) – \log_b(n)$
$\log_b(m^n) = n \cdot \log_b(m)$
You're asking questions and seeing if you can do things more efficiently -- that's what people who are good at math do.

I always forgot these when I first learned them. It might help to have a mnemonic. Anything that helps you remember is a mnemonic -- I find that getting an intuitive understanding or a feeling of why something is true helps me remember it.

Here's something that might help: think about $\log(n)$ as being the number of zeroes after $n$. (This is true if $n$ is a multiple of $10$ and the base of the log is $10$).


$\log_b (mn) = \log_b(m) + \log_b(n)$


If I multiply a number with three zeroes ($1,000$) by a number with two zeroes ($100$), how many zeroes will the answer have? Five: $100,000$.


$\log_b(m/n) = \log_b(m) – \log_b(n)$


Same deal as the previous, except with division. If I take $1,000$ and divide it by $100$, I get $10$. Three zeroes - two zeroes = one zero.


$\log_b(m^n) = n \cdot \log_b(m)$


If my number gets raised to the $n$-th power, how many more zeroes will it have? 
$$100^3 = 100 \cdot 100 \cdot 100 = 1,000,000$$

so $2$ zeroes to the $3$rd power is six zeroes. That's $n$ times as many.



Of course these laws also work for numbers that are not multiples of $10$. In fact, $10$ wasn't special to the above. But starting from the simplest case makes it easier to remember.

EDIT: Anything that helps you remember a good mnemonic, just want to stress that point. Something like "$\log$ takes the power down and puts it in the front" works perfectly well. I like the above because it helps me remember why the laws are true.
YES OF COURSE, people memorize those rules.  As far as "how" goes I don't know how to answer - you just do it.  One thing that ought to help is that it's all the same rule, really, because division is the inverse of multiplication and exponentiation is repeated multiplication.  Maybe it will help you to think of it that way.