How to solve $e^x = 2$ I know that $\ln(x)$ is the inverse of the exponential function $a^x$.
So I thought that
$$
e^x=2 \Leftrightarrow x = \ln(2)
$$
but my calculator says $x = \ln(2) + 2 i \pi n$, where $N \in \mathbb{Z}$. What have $e^x$ and $\ln(x)$ to do with the unit circle?

Euler's formula says that $$e^{i \theta}=\cos(\theta)+i\sin(\theta) \quad(*)$$
so let $\theta=2\pi n$ then $(*)$ becomes $e^{i2\pi n}=1$ 
so your calculator  is right
