Incoherence using Euler's formula

Question

Using the relation $\ e^{ix} = \cos(x) + i\sin(x)$ and substituting for $\ x = \pi$, we have the well-known Euler identity, $ e^{i\pi} = -1$. Substitute also for $ x = -\pi $, we have $ e^{-i\pi} = -1$. So we can say $ e^{i\pi} = e^{-i\pi}$ and taking $\ln$ on both sides, $i\pi=-i\pi$. There is clearly something wrong here. Can some help me to figure out what's wrong?

EDIT : I actually had a confusing output from my calculator while trying to solve the problem :

Answer 1

$e^{ix}=e^{iy}\implies e^{i(x-y)}=1=e^{2n\pi i}$ as $e^{2n\pi i}=\cos2n\pi+i\sin2n\pi=1$

$\implies x-y=2n\pi$ where $n$ is any integer

Answer 2

The logarithm is multi-valued. So we define a branch cut of the logarithm as a principal branch along an axis (or any line) so that we only use one cycle of phase of period $2 \pi$; this keeps that principal branch as a single-valued function.

Answer 3

In definition of $ln(z)$ for $z\in \mathbb{C}$ we have: $$ln(x+iy)=ln(|x|)+iArg(y)$$ Since $Arg$ can achive multi values by difference $2k\pi$ so $ln(x+iy)=ln(x+i(y+2k\pi))$.