Need help with a proof concerning zero-free holomorphic functions. Suppose $f(z)$ is holomorphic and zero-free in a simply connected domain, and that $\exists g(z)$ for which $f(z) =$ exp$(g(z))$.
The question I am answering is the following:
Let $t\neq 0$ be a complex number. Prove that $\exists h(z)$ holomorphic such that $f(z) = (h(z))^t$.
I see that the idea makes sense, but a nudge in the right direction would be appreciated.

Nudge: Remember that for real numbers $r>0$, you can define $r^t = \exp(t\ln r)$. Maybe you can do something similar for holomorphic functions?
