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 correspond to
any of the rules of exponents that you should have learnt before,
is the **change-of-base** formula:

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

*b*^{x}= 10^{x 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 a change of base formula to convert the others. That said, if the variable only appears in an equation once, then you can still always solve it by reversing all of the operations applied to the variable, which is often simpler.

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This web page was written between 2012 and 2018 by Toby Bartels, last edited on 2018 August 20. Toby reserves no legal rights to it.

The permanent URI of this web page
is
`http://tobybartels.name/MATH-1150/2018SU/loglaws/`

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