$\displaystyle x \in \mathbb{R}, e^{x} = e^{x \frac{2 \pi i}{2 \pi i}} = e^{2 \pi i \frac{x}{2 \pi i}} = (e^{2 \pi i})^{\frac{x}{2 \pi i}} = (cos(2 \pi) + i \text{ } sin(2 \pi))^{\frac{x}{2 \pi i}} = 1 $
$\displaystyle x \in \mathbb{R}, e^{x} = e^{x \frac{2 \pi i}{2 \pi i}} = e^{2 \pi i \frac{x}{2 \pi i}} = (e^{2 \pi i})^{\frac{x}{2 \pi i}} = (cos(2 \pi) + i \text{ } sin(2 \pi))^{\frac{x}{2 \pi i}} = 1 $
I think the error is in the step $\displaystyle e^{x\frac{2i\pi}{2i\pi}}=e^{2i\pi\frac{x}{2i\pi}$ For similar reasons to why this is incorrect:
$\displaystyle e^{i\pi}=-1$
$\displaystyle e^{2i\pi}=1$
$\displaystyle ln(e^{2i\pi})=ln(1)$
$\displaystyle 2i\pi=0$