Exponential on Real Numbers is Group Isomorphism

Theorem
Let $\struct {\R, +}$ be the additive group of real numbers.

Let $\struct {\R_{> 0}, \times}$ be the multiplicative group of positive real numbers.

Let $\exp: \struct {\R, +} \to \struct {\R_{> 0}, \times}$ be the mapping:
 * $x \mapsto \map \exp x$

where $\exp$ is the exponential function.

Then $\exp$ is a group isomorphism.