The exponential map from the additive reals to the positive reals

Example

Let $(\mathbb{R}, +)$ be the real numbers with addition, and let $(\mathbb{R}_{>0}, \times)$ be the positive real numbers with multiplication. The exponential function gives a map $\exp : \mathbb{R} \to \mathbb{R}_{>0}$ with $\exp(x + y) = \exp(x)\exp(y)$. Thus it is a homomorphism. Its kernel is $\{0\}$, because $\exp(x) = 1$ only for $x = 0$. Thus the map is injective. Each $y > 0$ is $\exp(\ln y)$, so the map is onto. Thus $\exp$ is an isomorphism.