Real Natural Function

The Real Natural Logarithm

Theorem: Injectivity of the Real Natural Logarithm

The real exponential function is injective on $(0;\infty )$.

PROOF

TODO

Definition: The Real Natural Logarithm

The real natural logarithm is the inverse function of the real exponential function on $(0;\infty )$.

NOTATION

$$\ln (x){\log }_{\mathrm{e}}(x)\log (x)$$

Properties

Addition and Subtraction Theorem for the Real Natural Logarithm

The real natural logarithm has the following property:

$$\ln (xy)=\ln x+\ln y\ln \left(\frac{x}{y}\right)=\ln x-\ln y$$

PROOF

TODO

Theorem: Continuity of the Real Natural Logarithm

The real batural logarithm is continuous.

PROOF

TODO

Theorem: Derivative of the Real Natural Logarithm

The real natural logarithm is differentiable everywhere where it is defined and

$$(\ln x{)}^{\prime }=\frac{1}{x}$$

PROOF

TODO

Theorem: Antiderivatives of the Real Natural Logarithm

The antiderivatives of the real natural logarithm are

$$\int \ln x=x\ln x-x+C$$

PROOF

TODO