Holomorphic logarithm of a nonvanishing function on a simply connected domain

Let \(\Omega\subseteq\CC\) be a simply connected open set and let \(f\) be holomorphic and nonvanishing on \(\Omega\). Then there is a holomorphic function \(G\) on \(\Omega\) such that \[ f(z)=e^{G(z)}\quad\text{for all } z\in\Omega. \]