Prove $\log_bf(x)$ is big-theta $\log f(x)$

Question

How can I prove that $\log_bf(x)$ is big-theta of $\log f(x)$ for any constant $b > 1$?

Answer 1

Hint: Note that $$\log_b(y)=\frac{\log y}{\log b}.$$

Remark: Slightly more generally, we have the change of base formula $$\log_b(y)=\frac{\log_a y}{\log_a b}.$$ This can be rewritten as $\log_a y=(\log_a b)(\log_b y)$, and then verified by raising $a$ to the power of each side.