Law of exponents: | Law of logarithms: |
---|---|

b^{0} = 1, |
log_{b} 1 = 0; |

b^{1} = b, |
log_{b} b = 1; |

b^{x + y} =
b^{x}b^{y}, |
log_{b} (uv) =
log_{b} u +
log_{b} v; |

b^{x − y} =
b^{x}/b^{y}, |
log_{b} (u/v) =
log_{b} u −
log_{b} v; |

b^{xy} =
(b^{y})^{x}, |
log_{b} (u^{x}) =
x log_{b} u; |

b^{y/x} =
^{x}√(b^{y}), |
log_{b} (^{x}√u) =
(log_{b} u)/x. |

Another important rule,
which doesn't directly correspond to any particular rule of exponents,
is the **change-of-base** formula:

- log
_{b}*u*= (log*u*)/(log*b*).

Each law of logarithms can be used in two directions: to break down the logarithm of a complicated expression into an expression involving simple logarithms, or to combine an expression into a single logarithm. When breaking down a logarithm, you may have to do some factoring.

To solve an equation involving logarithms with the same base, combine both sides into logarithms and drop the logs; to solve an equation involving variables in the exponents, take logarithms of both sides and break them down (after factoring as much as possible). If you have different bases in the same problem, pick one and use the change of base formula to convert the others.

Go back to the course homepage.

This web page was written between 2012 and 2016 by Toby Bartels, last edited on 2016 August 24. Toby reserves no legal rights to it.

The permanent URI of this web page
is
`http://tobybartels.name/MATH-1150/2016SU/loglaws/`

.