Could you tell me why the function $\mathbb{R}^2 \ni z \rightarrow e^z \in \mathbb{R}^2$, complex exponential, is not invertible globally. On a horizontal strip $[\ i y, \ i(y+2 \pi))$ its inverse is $\ln z$.

I found this example on [this forum](https://math.stackexchange.com/questions/39142/local-diffeomorphism), but there is no explanation if this fact. 
Or at least it is not explanatory enough for me to understand.

I would really appreciate all your insight.

Thank you.