The principal logarithm unwraps the upper half-plane into the strip $0<\Im w<\pi$.Suggest a slogan
Let \(S\coloneqq\theset{w\in\CC : 0<\Im w<\pi}=\RR\times i(0,\pi)\). The principal branch \(\Log\) of the logarithm restricts to a biholomorphism \[ \begin{aligned} F\colon \HH &\to S, \\ z &\mapsto \Log(z), \end{aligned} \] with inverse \(w\mapsto e^w\).
Every \(z\in\HH\) is \(z=re^{i\theta}\) with \(r>0\) and \(\theta\in(0,\pi)\), so \(\Log(z)=\log r+i\theta\in S\) and \(e^{\Log(z)}=z\). Conversely, for \(w=s+it\in S\), \(e^w=e^se^{it}\) has argument \(t\in(0,\pi)\), so \(e^w\in\HH\) and \(\Log(e^w)=s+it=w\). Thus \(F\) and \(w\mapsto e^w\) are mutually inverse, and both are holomorphic.