 Solution

Use the continuous argument $0<\arg z<2\pi$ on the positive-axis slit plane and defineThis analytic logarithm on the positive-axis slit plane is holomorphic, satisfies $e^{L(z)}=z$ and $L^{\prime }(z)=1/z$, and has $L(-1)=i\pi$. That value fixes the branch uniquely on the connected domain. For positive real $x$, the two boundary limits areThe jump, rather than a pole of the integrand, is what the keyhole argument exploits.