# Hva er regnereglene for logaritmer?

Som du ser av reglene under er alle logaritmereglene for tierlogaritmen $\mathrm{log}x$ og den naturlige logaritmen $\mathrm{ln}x$ like. Her kommer en oversikt som viser nettopp dette:

Briggske logaritmen:

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

Naturlige logaritmen:

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

Regel

### Logaritmereglerfortierlogaritmen

Første logaritmesetning:

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

Andre logaritmesetning:

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

Tredje logaritmesetning:

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

Regel

### Logaritmereglerfordennaturligelogaritmen

Første logaritmesetning

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

Andre logaritmesetning

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

Tredje logaritmesetning

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

Eksempel 1

Skriv $\mathrm{lg}{a}^{2}+\mathrm{lg}{b}^{2}-2\mathrm{lg}a$ så enkelt som mulig

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

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

Eksempel 2

Skriv $\mathrm{lg}ab+\mathrm{lg}{b}^{2}-\mathrm{lg}{a}^{2}b$ så enkelt som mulig

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

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

Eksempel 3

Skriv $\mathrm{lg}\frac{a}{b}-\mathrm{lg}\frac{2a}{{b}^{3}}$ enkelt som mulig

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

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

Eksempel 4

Skriv $\mathrm{lg}2x+\mathrm{lg}2-\mathrm{lg}\frac{2}{{x}^{2}}+\mathrm{lg}10$ enkelt som mulig

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

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

Eksempel 5

Skriv $\mathrm{lg}ab+\mathrm{lg}{b}^{2}-\mathrm{lg}{a}^{2}b$ så enkelt som mulig

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

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

Eksempel 6

Skriv $\mathrm{lg}\frac{a}{b}-\mathrm{lg}\frac{2a}{{b}^{3}}$ enkelt som mulig

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

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

Eksempel 7

Skriv $\mathrm{lg}2x+\mathrm{lg}2-\mathrm{lg}\frac{2}{{x}^{2}}+\mathrm{lg}10$ enkelt som mulig

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

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

Eksempel 8

Bruk logaritmesetningene til å forenkle uttrykket $\mathrm{ln}2x-\mathrm{ln}\phantom{\rule{-0.17em}{0ex}}\left(\frac{x}{2}\right)-4\mathrm{ln}x$

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

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

Eksempel 9

Bruk logaritmesetningene til å forenkle uttrykket $\mathrm{ln}2{x}^{3}-\mathrm{ln}\phantom{\rule{-0.17em}{0ex}}\left(\frac{3x}{2}\right)+\mathrm{ln}\phantom{\rule{-0.17em}{0ex}}{\left(3x\right)}^{2}$

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

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

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Hva er den naturlige logaritmen?

# Hva er regnereglene for logaritmer?

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