I’m trying to solve the following equation: $e^x = 2$. The solution, which I already know, is: $i(2 \pi n - i\ln(2))$ where $n \in \mathbb{Z}$.
The result makes perfect sense, however, I can’t seem to derive it from $re^{i\theta} = r(\cos(\theta) + i \sin(\theta))$.
If anyone has an idea on how to get to the result I’d greatly appreciate it.
Thanks
I am guessing what you are looking for.
$y=e^x=e^{u+iv}$, where $u$ and $v$ are real. $r=e^u$ and $\theta=v+2n\pi$.