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

Definition

The natural logarithm

$\ln :{ℝ}_{+}⟶ℝ,x⟼\ln x,$

is defined as the inverse function of the

real exponential function.

Theorem

The natural logarithm

$\ln :{ℝ}_{+}⟶ℝ,x⟼\ln x,$

is a

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

$\ln (x\cdot y)=\ln x+\ln y$

holds for all

$x,y\in {ℝ}_{+}$.

Proof

This follows from fact, fact, fact and fact  (6).

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