Suppose that $f, g$ are functions from the positive integers to the positive reals. Under what circumstances will $\log f(n)=O(\log g(n))$ imply $f(n)=O(g(n))$?
It's easy to see that this isn't always true: If $f(n)=3^n$ and $g(n)=2^n$ then we know that $3^n\notin O(2^n)$, but taking logs gives $\log3^n = n\log 3$ and $\log 2^n=n\log 2$ and it's obvious that $n\log 3=O(n\log 2)$.
The reason is also clear: If there exists a $c>0$ such that $\log f(n)\le c\,\log g(n)$ then all we can say is that $f(n)\le (g(n))^c$, which won't necessarily allow one to conclude $f(n)=O(g(n))$. When, though, can we correctly make the inference?