# What Are the Logarithm Rules?

As you can see below, all the logarithmic rules for the common logarithm $\mathrm{log}x$ and the natural logarithm $\mathrm{ln}x$ are the same:

Common logarithm:

$1{0}^{\mathrm{log}a}=a$

Natural logarithm:

${e}^{\mathrm{ln}a}=a$

This is true for any logarithm regardless of its base.

Rule

### TheLogarithmicRulesfortheCommonLogarithm

The first logarithmic rule:

 $\mathrm{log}\phantom{\rule{-0.17em}{0ex}}\left({a}^{x}\right)=x\mathrm{log}a$

The second logarithmic rule:

 $\mathrm{log}\phantom{\rule{-0.17em}{0ex}}\left(a\cdot b\right)=\mathrm{log}a+\mathrm{log}b$

The third logarithmic rule:

 $\mathrm{log}\phantom{\rule{-0.17em}{0ex}}\left(\frac{a}{b}\right)=\mathrm{log}a-\mathrm{log}b$

Rule

### TheLogarithmicRulesfortheNaturalLogarithm

The first logarithmic rule

 $\mathrm{ln}\phantom{\rule{-0.17em}{0ex}}\left({a}^{x}\right)=x\mathrm{ln}a$

The second logarithmic rule

 $\mathrm{ln}\phantom{\rule{-0.17em}{0ex}}\left(a\cdot b\right)=\mathrm{ln}a+\mathrm{ln}b$

The third logarithmic rule

 $\mathrm{ln}\phantom{\rule{-0.17em}{0ex}}\left(\frac{a}{b}\right)=\mathrm{ln}a-\mathrm{ln}b$

Example 1

Simplify $\mathrm{log}{a}^{2}+\mathrm{log}{b}^{2}-2\mathrm{log}a$

$\begin{array}{llll}\hfill \mathrm{log}{a}^{2}+\mathrm{log}{b}^{2}-2\mathrm{log}a& =2\mathrm{log}a+2\mathrm{log}b\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}-2\mathrm{log}a\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =2\mathrm{log}b\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill \mathrm{log}{a}^{2}+\mathrm{log}{b}^{2}-2\mathrm{log}a& =2\mathrm{log}a+2\mathrm{log}b-2\mathrm{log}a\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =2\mathrm{log}b\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Example 2

Simplify $\mathrm{log}ab+\mathrm{log}{b}^{2}-\mathrm{log}{a}^{2}b$

$\begin{array}{llll}\hfill \mathrm{log}ab+\mathrm{log}{b}^{2}-\mathrm{log}{a}^{2}b& =\mathrm{log}a+\mathrm{log}b+2\mathrm{log}b\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}-\phantom{\rule{-0.17em}{0ex}}\left(\mathrm{log}{a}^{2}+\mathrm{log}b\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\mathrm{log}a+3\mathrm{log}b-\mathrm{log}{a}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}-\mathrm{log}b\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\mathrm{log}a+2\mathrm{log}b\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}-2\mathrm{log}a\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =-\mathrm{log}a+2\mathrm{log}b\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill \mathrm{log}ab+\mathrm{log}{b}^{2}-\mathrm{log}{a}^{2}b& =\mathrm{log}a+\mathrm{log}b+2\mathrm{log}b-\phantom{\rule{-0.17em}{0ex}}\left(\mathrm{log}{a}^{2}+\mathrm{log}b\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\mathrm{log}a+3\mathrm{log}b-\mathrm{log}{a}^{2}-\mathrm{log}b\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\mathrm{log}a+2\mathrm{log}b-2\mathrm{log}a\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =-\mathrm{log}a+2\mathrm{log}b\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Example 3

Simplify $\mathrm{log}\frac{a}{b}-\mathrm{log}\frac{2a}{{b}^{3}}$

$\begin{array}{llll}\hfill \mathrm{log}\frac{a}{b}-\mathrm{log}\frac{2a}{{b}^{3}}& =\mathrm{log}a-\mathrm{log}b-\phantom{\rule{-0.17em}{0ex}}\left(\mathrm{log}2a-\mathrm{log}{b}^{3}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\mathrm{log}a-\mathrm{log}b-\left(\mathrm{log}2+\mathrm{log}a\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}-3\mathrm{log}b\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\mathrm{log}a-\mathrm{log}b-\mathrm{log}2-\mathrm{log}a\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}+3\mathrm{log}b\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =2\mathrm{log}b-\mathrm{log}2\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill \mathrm{log}\frac{a}{b}-\mathrm{log}\frac{2a}{{b}^{3}}& =\mathrm{log}a-\mathrm{log}b-\phantom{\rule{-0.17em}{0ex}}\left(\mathrm{log}2a-\mathrm{log}{b}^{3}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\mathrm{log}a-\mathrm{log}b-\left(\mathrm{log}2+\mathrm{log}a-3\mathrm{log}b\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\mathrm{log}a-\mathrm{log}b-\mathrm{log}2-\mathrm{log}a+3\mathrm{log}b\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =2\mathrm{log}b-\mathrm{log}2\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Example 4

Simplify $\mathrm{log}2x+\mathrm{log}2-\mathrm{log}\frac{2}{{x}^{2}}+\mathrm{log}10$

$\begin{array}{llll}\hfill \mathrm{log}2x+\mathrm{log}2-\mathrm{log}\frac{2}{{x}^{2}}+\mathrm{log}10& =\mathrm{log}2+\mathrm{log}x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{1em}{0ex}}+\mathrm{log}2-\left(\mathrm{log}2\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{1em}{0ex}}-\mathrm{log}{x}^{2}\right)+1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =2\mathrm{log}2+\mathrm{log}x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{1em}{0ex}}-\mathrm{log}2+\mathrm{log}{x}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{1em}{0ex}}+1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\mathrm{log}2+\mathrm{log}x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{1em}{0ex}}+2\mathrm{log}x+1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\mathrm{log}2+3\mathrm{log}x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{1em}{0ex}}+1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill \mathrm{log}2x+\mathrm{log}2-\mathrm{log}\frac{2}{{x}^{2}}+\mathrm{log}10& =\mathrm{log}2+\mathrm{log}x+\mathrm{log}2-\phantom{\rule{-0.17em}{0ex}}\left(\mathrm{log}2-\mathrm{log}{x}^{2}\right)+1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =2\mathrm{log}2+\mathrm{log}x-\mathrm{log}2+\mathrm{log}{x}^{2}+1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\mathrm{log}2+\mathrm{log}x+2\mathrm{log}x+1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\mathrm{log}2+3\mathrm{log}x+1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Example 5

Use the logarithmic rules to simplify $\mathrm{ln}2x-\mathrm{ln}\phantom{\rule{-0.17em}{0ex}}\left(\frac{x}{2}\right)-4\mathrm{ln}x$

$\begin{array}{llll}\hfill & \phantom{=}\mathrm{ln}2x-\mathrm{ln}\phantom{\rule{-0.17em}{0ex}}\left(\frac{x}{2}\right)-4\mathrm{ln}x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\mathrm{ln}2+\mathrm{ln}x-\phantom{\rule{-0.17em}{0ex}}\left(\mathrm{ln}x-\mathrm{ln}2\right)-4\mathrm{ln}x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\mathrm{ln}2+\mathrm{ln}x-\mathrm{ln}x+\mathrm{ln}2-4\mathrm{ln}x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =2\mathrm{ln}2-4\mathrm{ln}x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill \mathrm{ln}2x-\mathrm{ln}\phantom{\rule{-0.17em}{0ex}}\left(\frac{x}{2}\right)-4\mathrm{ln}x& =\mathrm{ln}2+\mathrm{ln}x-\phantom{\rule{-0.17em}{0ex}}\left(\mathrm{ln}x-\mathrm{ln}2\right)-4\mathrm{ln}x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\mathrm{ln}2+\mathrm{ln}x-\mathrm{ln}x+\mathrm{ln}2-4\mathrm{ln}x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =2\mathrm{ln}2-4\mathrm{ln}x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Example 6

Use the logarithmic rules to simplify $\mathrm{ln}2{x}^{3}-\mathrm{ln}\phantom{\rule{-0.17em}{0ex}}\left(\frac{3x}{2}\right)+\mathrm{ln}\phantom{\rule{-0.17em}{0ex}}{\left(3x\right)}^{2}$

$\begin{array}{llll}\hfill & \phantom{=}\mathrm{ln}2{x}^{3}-\mathrm{ln}\phantom{\rule{-0.17em}{0ex}}\left(\frac{3x}{2}\right)+\mathrm{ln}\phantom{\rule{-0.17em}{0ex}}{\left(3x\right)}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\mathrm{ln}2+\mathrm{ln}{x}^{3}-\phantom{\rule{-0.17em}{0ex}}\left(\mathrm{ln}3x-\mathrm{ln}2\right)+\mathrm{ln}{3}^{2}{x}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\mathrm{ln}2+3\mathrm{ln}x-\phantom{\rule{-0.17em}{0ex}}\left(\mathrm{ln}3+\mathrm{ln}x-\mathrm{ln}2\right)+\mathrm{ln}{3}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}+\mathrm{ln}{x}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\mathrm{ln}2+3\mathrm{ln}x-\mathrm{ln}3-\mathrm{ln}x+\mathrm{ln}2+2\mathrm{ln}3\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}+2\mathrm{ln}x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =2\mathrm{ln}2+4\mathrm{ln}x+\mathrm{ln}3\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill \mathrm{ln}2{x}^{3}-\mathrm{ln}\phantom{\rule{-0.17em}{0ex}}\left(\frac{3x}{2}\right)+\mathrm{ln}\phantom{\rule{-0.17em}{0ex}}{\left(3x\right)}^{2}& =\mathrm{ln}2+\mathrm{ln}{x}^{3}-\phantom{\rule{-0.17em}{0ex}}\left(\mathrm{ln}3x-\mathrm{ln}2\right)+\mathrm{ln}{3}^{2}{x}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\mathrm{ln}2+3\mathrm{ln}x-\phantom{\rule{-0.17em}{0ex}}\left(\mathrm{ln}3+\mathrm{ln}x-\mathrm{ln}2\right)+\mathrm{ln}{3}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}+\mathrm{ln}{x}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\mathrm{ln}2+3\mathrm{ln}x-\mathrm{ln}3-\mathrm{ln}x+\mathrm{ln}2+2\mathrm{ln}3\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}+2\mathrm{ln}x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =2\mathrm{ln}2+4\mathrm{ln}x+\mathrm{ln}3\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$