Logarithmic functions > Properties
12345Properties

## Theory

Definition of a logarithm:
${g}^{x}=y$ is equivalent to $x={}^{g}log\left(y\right)$ if $0 or $g>1$ and if $y>0$.

Definition formulas:
It follows from the definition of a logarithm that: ${}^{g}log\left({g}^{x}\right)=x$ and ${g}^{{}^{g}log\left(y\right)}=y$.

Properties of logarithms:
If $0 or $g>1$ and if $a>0$ and $b>0$ the following properties hold:

• ${}^{g}log\left(a\right)+{}^{g}log\left(b\right)={}^{g}log\left(a\cdot b\right)$

• ${}^{g}log\left(a\right)-{}^{g}log\left(b\right)={}^{g}log\left(\frac{a}{b}\right)$

• $p\cdot {}^{g}log\left(a\right)={}^{g}log\left({a}^{p}\right)$

Changing the base:
In order to be able to work with a desired base (the log-button of your calculator always uses 10) you need to be able to change an existing base.
From the properties of logarithms you can derive: ${}^{g}log\left(a\right)={}^{p}log\left(a\right)/{}^{p}log\left(g\right)$.
In this way you can determine logarithms with your calculator and/or enter them as a function. Note that modern calculators sometimes allow you to choose the base of the logarithm. You then need to use the American notation ${log}_{g}\left(x\right)$, where the base is written in a different position.