When is the big-O relation preserved under exponentiation?
It is at least sufficient to have $\log f(n) \in o(\log g(n))$. This gives us that for sufficiently large $N$, we have $\log f(n) \leq \log g(n)$ for all $n &gt; N$.

Hence immediately we have that for $c=1, n &gt; N$, $f(n) \leq c\cdot g(n)$ and ergo $f(n) \in O(g(n))$.

This also suggests the condition that if we have that $\log f(n) \leq c\cdot \log (g(n))$ where $c \leq 1$ for sufficiently large $n$, then we also get $f(n) \in O(g(n))$.