## Algebra## Exponents and Logarithms |

## What are the rules for combining logarithms? |

There are certain rules for combining logarithms. In the following cases, let *a* be a positive number that does not equal 0; *n* be a real number; and *u* and *v* be positive real numbers:

Logarithmic Rule 1: log_{a}(*uv*) = log_{a}(*u*) + log_{a}(*v*)

Logarithmic Rule 2: log_{a}(*u/v*) = log_{a}(*u*) - log_{a}(*v*)

Logarithmic Rule 3: log_{a}(*u*)^{n} = *n*log_{a}(*u*)

This can be expressed as follows: In rule one, multiplication inside the log is turned into addition outside the log (and vice versa); in rule two, division inside the log is turned into subtraction outside the log (and vice versa); and in rule three, an exponent on anything inside the log can be moved to the front of the log as a multiplier (and vice versa). But remember, these rules only apply if the bases are the same. For example, because the bases are not the same in log_{a}(*u*) + log_{b}(*v*), this expression can’t be simplified.