= Solution
Use the continuous argument $0<\arg z<2\pi$ on the positive-axis slit plane and define
$$
\boxed{L(z)=\log|z|+i\arg z.}
$$
This <analytic logarithm on the positive-axis slit plane> is holomorphic, satisfies $e^{L(z)}=z$ and $L'(z)=1/z$, and has $L(-1)=i\pi$. That value fixes the branch uniquely on the connected domain. For positive real $x$, the two boundary limits are
$$
L(x+i0)=\log x,\qquad L(x-i0)=\log x+2\pi i.
$$
The jump, rather than a pole of the integrand, is what the keyhole argument exploits.
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