Logarithm tricks

This article/section is a stub — probably a pile of half-sorted notes and is probably a first version, is not well-checked, so may have incorrect bits. (Feel free to ignore, or tell me)

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:

logw v = ( logh v ) / ( logh 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

log2(256) = ( log10(256) ) / ( log10(2) ) ≈ 2.40824 / 0.30103 ≈ 8

...or are working with base e

log2(256) = ( loge(256) ) / ( loge(2) ) ≈ 5.54518  / 0.69315 ≈ 8

addition is multiplication

logb(x*y) = logb x + logb y

subtraction is division

logb(x/y) = logb x - logb y

Decibels

Decibels are logarithms applied to ratios of power.

Which turns out to be rather useful idea.

Log probabilities

https://en.wikipedia.org/wiki/Log_probability