Natural logarithm/Inverse function to real exponential function/Introduction/Section

Definition

The natural logarithm

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

is defined as the inverse function of the

real exponential function.

Theorem

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

This follows from

and

$\Box$