= Solution
Write $z=re^{i\theta}$ with $r>0$ and $0\leq\theta<2\pi$, and write $w=u+iv$. Since
$$
e^w=e^ue^{iv}=re^{i\theta},
$$
we have
$$
u=\log r,
\qquad
v=\theta+2\pi n,
\qquad n\in\mathbb Z.
$$
Thus the inverse of the exponential is the infinitely valued <complex logarithm>
$$
\boxed{w(z)=\log r+i(\theta+2\pi n),\qquad n\in\mathbb Z.}
$$
For the angle convention in the question, its principal value is
$$
\boxed{\operatorname{Log}z=\log r+i\theta,\qquad0\leq\theta<2\pi.}
$$
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