the periodicity of the complex exponential

The real exponential e^x never repeats: it climbs forever and is one-to-one. So a complex newcomer naturally expects e^z to be one-to-one as well. It is not, and the reason is the single most important structural fact about the exponential.

Because e^z = e^x (cos y + i sin y) and cosine and sine have period 2 pi, adding 2 pi i to the exponent changes nothing: e^(z + 2 pi i) = e^z. More precisely e^z is periodic with period 2 pi i, and these are the only periods (they are exactly the multiples 2 pi i k for integers k). So the exponential maps every horizontal strip of height 2 pi, say the strip -pi < y <= pi, one-to-one onto the whole plane minus the origin, and then repeats the same picture in the strip above and below, forever. Two inputs give the same output exactly when they differ by an integer multiple of 2 pi i.

This periodicity is the source of multi-valuedness downstream. Since infinitely many z share one value e^z, undoing the exponential cannot give a single answer: that is precisely why log w has infinitely many values differing by 2 pi i, and why we must choose a branch. The 2 pi i here and the 2 pi ambiguity in the argument of a complex number are the same phenomenon.