Simplifying Logarithmic Expressions
Properties of Logarithms
The properties of logarithms are analogous to the properties of exponents .
${\mathrm{log}}_{b}b=1$ for any $b$ 
Since
${b}^{1}=b$

${\mathrm{log}}_{b}1=0$ for any $b$ 
Since
${b}^{0}=1$

${\mathrm{log}}_{b}0$ is undefined for all $b$ 
Since there is no
$x$
for which
${b}^{x}=0$

${\mathrm{log}}_{b}x$ is undefined if $x$ is negative 
It may seem like ${\mathrm{log}}_{2}\left(8\right)$ should equal $3$ , since ${\left(2\right)}^{3}=8$ . But on the other hand, ${\mathrm{log}}_{2}\left(4\right)$ doesn't mean anything: the equation ${\left(2\right)}^{x}=4$ has no solution. 
${\mathrm{log}}_{b}xy={\mathrm{log}}_{b}x+{\mathrm{log}}_{b}y$

Since
${b}^{m}\cdot {b}^{n}={b}^{m+n}$

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