Group isomorphism/Real exponential function/Example

We consider the additive group of the real numbers, that is ${}(\mathbb {R} ,0,+)$, and the multiplicative group of the positive real numbers, thus ${}(\mathbb {R} _{+},1,\cdot )$. Then the exponential function

$\exp \colon \mathbb {R} \longrightarrow \mathbb {R} _{+},x\longmapsto \exp(x),$

is a group isomorphism. This rests on basic analytic properties of the exponential function. The homomorphism property is just a reformulation of the functional equation

${}\exp(x+y)=e^{x+y}=e^{x}e^{y}=\exp(x)\exp(y)\,.$

The injectivity of the mapping follows from the strict monotonicity, the surjectivity follows from the Intermediate value theorem. The inverse mapping is the natural logarithm, which is also a group isomorphism.