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

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

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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.

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Hey Andre, thanks so much for all the help! How are you so good at math and proofs? xD I want to learn your ways xD – user43564 Oct 4 '12 at 6:23
When you have been doing it as long as I have, you may be much better than I am. – André Nicolas Oct 4 '12 at 6:25
Nooo I don't know about that, you are a beast. I'm struggling in my Discrete mathematics and probability theory class and don't know how to become better at this kind of stuff, do you have any advice? – user43564 Oct 4 '12 at 6:26
Practice, practice, practice. – André Nicolas Oct 4 '12 at 6:30
Alright thank you Andre, I appreciate it a lot! – user43564 Oct 4 '12 at 6:34