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.