Exponential on Real Numbers is Group Isomorphism

Theorem
Let $\left({\R, +}\right)$ be the additive group of real numbers.

Let $\left({\R_{> 0}, \times}\right)$ be the multiplicative group of positive real numbers.

Let $\exp: \left({\R, +}\right) \to \left({\R_{> 0}, \times}\right)$ be the mapping:
 * $x \mapsto \exp \left({x}\right)$

where $\exp$ is the exponential function.

Then $\exp$ is a group isomorphism.