the multi-valuedness of the logarithm

Saying log z is multi-valued sounds like a defect, as if we could not decide on an answer. It is better understood as honest bookkeeping: the exponential genuinely sends many inputs to the same output, so undoing it genuinely produces many outputs, and the logarithm is simply reporting all of them.

Concretely, log z = ln|z| + i(arg z + 2 pi k) over all integers k. Fix a non-zero z. As k runs through the integers you get a vertical ladder of complex numbers all with the same real part ln|z| and imaginary parts spaced exactly 2 pi apart. Each rung is a legitimate logarithm: exponentiating any of them returns z. The deeper way to picture it: as a point traces a loop once counterclockwise around the origin, its argument increases by 2 pi, so a continuously chosen value of log z increases by 2 pi i and does not come back to where it started. The values do not just sit there as a list; they are reached by winding around the branch point at 0.

This multi-valuedness is not a quirk of the logarithm only. It is the seed of branch points, branch cuts, and Riemann surfaces, and it is inherited by everything built on the logarithm, complex powers z^a most of all. The practical response is always the same: to compute or differentiate, you choose a single-valued branch on a cut plane and stay on it; to understand the global structure, you let the value wind. Both viewpoints are correct, and the ambiguity is always a multiple of 2 pi i.