Group isomorphism/Real exponential function/Example

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

$\exp :ℝ⟶{ℝ}_{+},x⟼\exp (x),$

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.