Real Logarithms

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 of the real exponential function on $(0;\infty )$.

NOTATION

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

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 natural logarithm is continuous.

PROOF

TODO

Theorem: Derivative of the Real Natural Logarithm

The real natural logarithm is differentiable with

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

PROOF

TODO

Theorem: Antiderivatives of the Real Natural Logarithm

The antiderivatives of the real natural logarithm are given by

$$\int \ln x=x\ln x-x+\text{const}$$

PROOF

TODO

Real Logarithms

Definition: Real Logarithm

Let $b\in {\mathbb{R}}_{>0}$ and $b\ne 1$.

The logarithm with base $b$ is the function ${\log }_b:{\mathbb{R}}_{>0}\to \mathbb{R}$ defined using the real natural logarithm as

$${\log }_b(x)\stackrel{\text{def}}{=}\frac{\ln (x)}{\ln (b)}$$

for each $x>0$.

NOTATION

We can also write ${\log }_{b}x$.

Theorem: Logarithm as Solution to Equations

If $b\in {\mathbb{R}}_{>0}$ and $b\ne 1$ and $y>0$, then the equation

$$b^x=y$$

has the only solution

$$x={\log }_{b}y$$

PROOF

TODO

Theorem: Multiplication $\leftrightarrow$ Addition

Logarithms can be used to turn products into sums and viceversa:

$${\log }_b(a_1\times \cdots \times a_n)={\log }_b(a_1)+\cdots +{\log }_b(a_n)$$

PROOF

TODO

Theorem: Division $\leftrightarrow$ Subtraction

Logarithms can be used to turn division into subtraction and vice versa:

$${\log }_b\left(\frac{a}{c}\right)={\log }_b(a)-{\log }_b(c)$$

PROOF

TODO

Theorem: Exponentiation $\leftrightarrow$ Multiplication

Logarithms can be used to turn multiplication into exponentiation and vice versa:

$${\log }_b(a^c)=c\cdot {\log }_b(a)$$

PROOF

TODO

Theorem: Base Change

Let $a>0$ and $b_1,b_2>0$ with $b_1\ne 1$ and $b_2\ne 1$.

We can turn the real logarithm with base $b_1$ to real logarithms with base $b_2$ as follows:

$${\log }_{b_1}a=\frac{{\log }_{b_2}a}{{\log }_{b_2}{b}_1}$$

PROOF

TODO