e^(pi i)

e^(pi i) = x
ln (e^(pi i)) = ln x
(pi i)(lne) = ln x
pi i = ln x

domain of: x > 0

I must've missed something...

The "domain" of a real function refers to which real x-values produce real y-values. pi i isn't a real value, so naturally it isn't in the real domain of ln.

The equation follows from the identity e^(ix) = cos(x) + i sin(x), for which proofs are all over the Internet. The function's called cis(x) and has some interesting properties. From its definition it's pretty apparent that cis(pi) = -1, among other things.

Reference: http://mathworld.wolfram.com/EulerFormula.html