We were doing a quick review of undergrad topics the other day in my grad Algorithms class and the professor asked a simple question: Which grows faster, $2^n$ or $3^n$? Everyone was quick to agree that $3^n$ grew faster but I opened my big mouth and said "Obviously they grow at the same rate because if you take the log of both sides they have the same asymptotic behavior." I see the mistake I made in relating the functions themselves to the ones after taking the logs but that brought up this question:
Why does taking the log of functions with different asymptotic behaviors bring them into a set of functions with the same asymptotic behavior? Does this apply to other function groups besides exponentials with different bases?
It's probably a dumb questions and I'm missing something extremely obvious. My prof said something like "logs tend to wash out or flatten functions when you use them so be careful" but I want something a bit more rigorous.
Obviously taking the log of a function affects the growth rate. I realize my title is misleading abit. My question might be better stated as:
$2^n$ and $3^n$ have different asymptotic behaviors. However, after taking the log of both functions, the new functions now have the same asymptotic behavior. Why?