= Solution
Choose the continuous argument with values in $(-\pi/2,3\pi/2)$ and define
$$
\boxed{\operatorname{Log}z=\log|z|+i\operatorname{Arg}z,\qquad -\frac\pi2<\operatorname{Arg}z<\frac{3\pi}2.}
$$
Its <branch cut> is the nonpositive imaginary axis, including zero, exactly the removed ray. Locally away from that ray, the <Cauchy-Riemann equations> or the local inverse of the exponential show that it is a <holomorphic logarithm>, with <derivative> $1/z$ and $e^{\operatorname{Log}z}=z$. The chosen interval fixes the additive ambiguity by integer multiples of $2\pi i$.
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