Natural logarithm/Functional equation/Bijection/Continuity/Monotonicity/Fact

The natural logarithm

$\ln \colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,x\longmapsto \ln x,$

is a

continuous strictly increasing function, which defines a bijection between ${}\mathbb {R} _{+}$ and ${}\mathbb {R}$. Moreover, the functional equation

${}\ln(x\cdot y)=\ln x+\ln y\,$

holds for all

${}x,y\in \mathbb {R} _{+}$.

Proof, Write another proof