Let \(\Log\) be the principal branch of the logarithm. The map \[ F\colon\HH=\ts{z\st\Im z>0}\to\ts{w\st0<\Im w<\pi},\qquad F(z)=\Log z, \] is a biholomorphism, with inverse \(w\mapsto e^w\).
For \(z\in\HH\) write \(z=re^{i\theta}\) with \(r>0\) and \(\theta\in(0,\pi)\); then \(\Log z=\ln r+i\theta\). Since \(r\mapsto\ln r\) is a bijection \((0,\infty)\to\RR\), \(F\) is a bijection onto the strip, and \(e^{\ln r+i\theta}=re^{i\theta}\) gives the holomorphic inverse.
The map extends continuously to the boundary of \(\HH\) minus \(0\), with the argument \(\pi\) on the negative real axis. As \(x\) increases from \(-\infty\) to \(0\), \(F(x)=\ln\abs x+i\pi\) moves from \(+\infty+i\pi\) to \(-\infty+i\pi\) along the line \(\Im w=\pi\). As \(x\) increases from \(0\) to \(\infty\), \(F(x)=\ln x\) moves from \(-\infty\) to \(+\infty\) along \(\RR\).