# Math Help - Failure of power and logarithm identities

1. ## Failure of power and logarithm identities

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 $(e^a)^b \neq e^{ab}$ 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.

2. They are true as long as you are dealing with real numbers!

3. Well I'm mostly concerned about using the identities for complex numbers. For example, the complex definition of $z^a$ is $z^a \equiv e^{a\log z}$. How can that be true when $\log z^a \neq a\log z$ when a and z are complex numbers?