logarithmic rules with functions

Question

Do the logarithmic rules work when taking logs of functions as opposed to numbers?

i.e. suppose $f$ is a function and $n$ is a real number, is $\log (f(x)^n) = n · \log(f(x))$?

Answer 1

Yes it will work.

$$\log(f(x)^n) = \log(f(x)\times f(x) ... \times f(x)$$ $$= \log(f(x)) +\log(f(x)) + ... +\log(f(x))$$ $$=n \times \log(f(x))$$

Answer 2

Yes, of course.

Because $f(x)$ is still a number for any $x$.

It even works on expressions: for example $\log((x^2+3)^9)=9\log(x^2+3).$