# 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 ${b}^{m}\cdot {b}^{n}={b}^{m+n}$ ${\mathrm{log}}_{b}\frac{x}{y}={\mathrm{log}}_{b}x-{\mathrm{log}}_{b}y$