I was tinkering with Euler's Identity and I come to wonder if it was possible to derive $\pi$ from it.
I know $\pi$ can't be expressed as a fraction of two rational numbers but neither $i$ nor $e$ is rational.
$e^{\pi i} = -1$ (square both sides)
$e^{2\pi i} = 1$ (logarithm both sides)
$2\pi * i * \log e = \log 1 = 0$
This is how far I got before meeting a contradiction as this the left side equals roughly $9.1 i$. Is it even possible to derive $\pi$ from Euler's Identity and where have I messed up?