Exponentiation - Wikipedia, the free encyclopedia

I was reading a bit on Wikipedia about the complex exponential function when I found the above link. I'm a bit confused about it, especially the fact that $\displaystyle (e^a)^b \neq e^{ab}$ when a or b are complex numbers. A lot of definitions and proofs seem to use those identities, so I'm curious to know how you're supposed to use them when the identity is not always true.