Why does this work, and why is it wrong? I have devised a "proof" that $i=0$. Obviously it can't be true, but I can't see why it is wrong. I've been thinking about this for a while, and my friend and I are very confused. What is wrong with the "proof" below?
$e^{i\pi} = -1$
$e^{2i\pi} = 1$
$2i\pi = ln(1)$
$i = \frac{0}{2\pi}$
$i = 0$

Now that you have proved in the second line $e^{2i\pi} = 1=e^0$, so the complex exponential function is not injective. Thus you can not define a complex log function as its inverse function.
