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:

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

where $\exp$ denotes the complex exponential.