Prove that the exponential $\exp z$ is not zero for any $z \in \Bbb C$ How can the following been proved?

$$
\exp(z)\neq0, z\in\mathbb{C}
$$

I tried it a few times, but I failed. Probably it is extremly simple.
If a draw the unit circle and then a complex number $\exp(a+ib)=\exp(a)\exp(ib)$ then it is obvious that this expression is only $0$ if $\exp(a)$ equals zero, but $\exp(a),a\in\mathbb{R}$ is never zero. This seems not to be very robust.
Thank you

If you know that $\exp(z+w)=\exp(z) \exp(w)$, then  $\exp(z)\ne 0$ follows from
$$1=\exp(0)=\exp(z-z)=\exp(z)\exp(-z)$$
