The complex log function is a "branching function" [1] with an infinite number of branches. For comparison, the square root function has two branches.

sqrt(x^2) has two values, +x and -x.

ln(e^x) has an infinite number of values, x + 2nπi (for any integer n).

2πi = 0 + 2nπi for some n. (You can repeat this for, say, e^4πi, e^989781123972πi, e^-091784πi, etc. Those are all equal to 1.)

[1] http://en.wikipedia.org/wiki/Complex_logarithm