Why is $\log(e^z)\ne z$ in complex analysis? I always know that the natural logarithm $\log_e$ is the inverse of the exponential function $e$, but to my surprise when reading elementary functions in complex analysis, I discover that that is not always true. $$\log(e^z)=z+2k\pi i$$
My question here is that, why is the above expression true? 
 A: The logarithm is not well-defined, because a function must be one-to-one to have an inverse. Since $e^{z}=e^{z+2\pi i}$, the exponential function is not one-to-one. We sometimes define a complex logarithm function by making a choice, for example we could insist that the imaginary part of $\log(z)$ is in the interval $[0,2\pi)$ for all $z$. But for any such choice, $\log(e^z)$ will differ from $z$ by a multiple of $2\pi i$ for most values of $z$. This is unavoidable, since $e^{z}=e^{z+2\pi i}$ and thus $\log(e^z)=\log(e^{z+2\pi i})$.
A: The better question perhaps, is why is $e^z$ periodic?
Well this has somewhat of a geometric interpretation, and for simplicity lets stick with a complex number of the form $i \theta$. Then:
$$e^{i \theta} = \left( \cos \theta + i \sin \theta \right)$$
and you can think of this as identifying the point $(1,\theta)$ on the unit circle in the complex plane (this is polar coordinates). But then of course if you add $2 \pi$ to this angle, you just get back to the same point on the circle. Thus the complex exponential function cannot be injective.
Thus if you want the logarithm of $w$ to be a complex number $z$ such that $e^z=w$, then this will also carry a periodicity. 
