On a simply connected region avoiding $0$, the complex logarithm has a single-valued holomorphic branch.Suggest a slogan
Let \(\Omega\subseteq\CC\) be a simply connected region with \(1\in\Omega\) and \(0\notin\Omega\). Then there exists a holomorphic function \(F\colon\Omega\to\CC\) such that
\(e^{F(z)}=z\) for all \(z\in\Omega\), and
\(F(r)=\log r\) for all real \(r\) in some neighborhood of \(1\) in \(\Omega\).