Let $U \subset \mathbb{C}$ \ {0} is non-empty, open, connected subset and $F: U \to \mathbb{C}$ is a continous function satisfying $e^{F(z)} = z$.
I have to show that a branch of the logarithm on U, if it exists, is unique only up to addition of a constant in $2\pi i \mathbb{Z} \in \mathbb{C}$.
I don't really have an idea how to show this. Thank you in advance.
Hint: If $F_1$ and $F_2$ are two such branches of the logarithm, then$$e^{F_1(z)-F_2(z)}=\frac{e^{F_1(z)}}{e^{F_2(z)}}=\frac zz=1.$$