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.