How to prove $\log(\exp(z))\neq z$ if $z\in \mathbb{C}$

Question

I have on my notes $5$ equalities regarding complex numbers that I have to prove. And I'm not sure on how to proceed with this one.

What's the best strategy to prove $\log(\exp(z))\neq z$ if $z\in \mathbb{C}$?

Thanks for your time

Answer 1

The best strategy depends on the way you have defined $\ln$. If, say, $\ln(\rho e^{i\theta})=\ln(\rho)+i\theta$ when $\theta\in[-\pi,\pi)$, then you can observe that $\ln(e^{2\pi i})=0\neq2\pi i$.

Answer 2

(I'm assuming you want to show the negation of "$\forall z, \log(\exp(z)) = z$", instead of "$\forall z, \log(\exp(z)) \neq z$", which has a chance of being false)

$\exp$ is not injective so it can't be inverted. For example this would imply $2i\pi = \log(\exp(2i\pi)) = \log(1) = \log(\exp(0)) = 0$