Holomorphic logarithm of a nowhere zero holomorphic function

Definition

Suppose $U\subset \mathbb {C}$ is an open subset and $f:U\to \mathbb {C} ^{*}$ is a nowhere zero holomorphic function on $U$. A holomorphic logarithm of $f$ is a holomorphic function $g:U\to \mathbb {C}$ such that:

$f(z)=\exp(g(z))\ \forall \ z\in U$

where $\exp$ denotes the complex exponential.