I am looking for approximations for $\exp \left( \log^{f(x)} g(x) \right)$.
Obviously if $f(x) = 1$ this would simplify to $g(x)$. Is there anything else t do when $f(x)\neq 1$? Any upper-bound or lower-bound approximations should be fine.
Clarification on the notation: $\log^i x = ( \log x )^i$
In theoretical computer science the following is often relevant:
If $c$ is a constant greater than $1$, $\exp (\log g(x) ^ c)$ is asymptotically greater than $p(g(x))$ for any polynomial $p$.