# Are exponential functions necessarily one to one?

If $a , b \in \mathbb{R}$ and $a , b > 0$, then $f \left(x\right) = a {b}^{x}$ is one-one from $\mathbb{R}$ onto $\left(0 , \infty\right)$
As a Complex valued function from $\mathbb{C} \to \mathbb{C}$ \ $\left\{0\right\}$, the exponential function $\exp \left(z\right) = {e}^{z}$ is many to one.
For example, ${e}^{2 k \pi i} = 1$ for all $k \in \mathbb{Z}$