Solution
= Solution
In the <upper half-plane>, the argument lies strictly between zero and $\pi$, while $\log|z|$ can be any real number. Conversely, if $w=s+it$ with $0<t<\pi$, then $z=e^w$ is in the <upper half-plane> and $L(z)=w$.
$$
\boxed{L(H)=\{w\in\mathbb C:0<\operatorname{Im}w<\pi\}.}
$$
Thus the <principal complex logarithm> is a <conformal bijection> from the <upper half-plane> onto that horizontal strip, with inverse the <exponential function> restricted to the strip.