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Fix $z_0\ne 0,1$ and take a small disk about it that avoids zero and one. There is a holomorphic logarithm $L$ on this disk. The function $L(z)/(2\pi i)$ cannot be an integer there: if it were an integer, exponentiation would give $z=1$. Thus $f(L(z)/(2\pi i))$ is holomorphic on the disk.

Where the prescribed principal logarithm and $L$ are both evaluated, they differ by $2\pi ik$ for an integer $k$. By the period-one identity, their compositions with $f$ coincide. This remains true on either side of the principal branch cut, and at a point on that cut one may instead choose the logarithm with argument between zero and $2\pi$. The local holomorphic expressions therefore agree with the stated pointwise definition and with each other. This is periodic holomorphic descent through the exponential map, provingThe principal logarithm itself is not analytic across its cut; periodicity of the outer function removes that obstruction.