Definition of a logarithm:
${g}^{x}=y$ is equivalent to
$x={}^{g}log\left(y\right)$ if
$0<g<1$ 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
$0<g<1$ or
$g>1$ and if
$a>0$ and
$b>0$ the following properties hold:
${}^{g}log\left(a\right)+{}^{g}log\left(b\right)={}^{g}log(a\cdot b)$
${}^{g}log\left(a\right)-{}^{g}log\left(b\right)={}^{g}log\left(\frac{a}{b}\right)$
$p\cdot {}^{g}log\left(a\right)={}^{g}log\left({a}^{p}\right)$
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.