# Logarithm tricks

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### properties of logs

#### Change of log base

If you want to work with an arbitrary base, but you have a log function of which you cannot change the base (e.g. on a calculator), then a useful property is:

log_{w}v = ( log_{h}v ) / ( log_{h}w )

where

- w is the base you want
- h is the base you have (which you don't even have to know the value of, just has to be consistent just has to be consistent between the two uses)
- v is the value you want the log
_{w}of

Say you want the log-base-2 of 256

- ...but are working with base 10

log_{2}(256) = ( log_{10}(256) ) / ( log_{10}(2) ) ≈ 2.40824 / 0.30103 ≈ 8

- ...or are working with base e

log_{2}(256) = ( log_{e}(256) ) / ( log_{e}(2) ) ≈ 5.54518 / 0.69315 ≈ 8

#### addition is multiplication

log_{b}(x*y) = log_{b}x + log_{b}y

#### subtraction is division

log_{b}(x/y) = log_{b}x - log_{b}y

#### other

### Decibels

Decibels are logarithms applied to ratios of power.

Which turns out to be rather useful idea.