Base change formula - Logarithms

Meta: This post is about the base change formula for logarithms. It helps understand how to change the base of a logarithm to another base.

The formula

`log_m(a) = (log_n(a))/(log_n(m))`

An example

The base of the logarithm `log_10(5)` is 10. 
Let us convert the base of `log_10(5)` to e.
  • Original logarithm value = 5
  • Original base = 10
  • New base = e
Applying the base change formula,
`log_10(5) = (log_e(5))/(log_e(10))` 
Thus the base of the logarithm `log_10(5)` is changed from 10 to e.


The base change formula changes the base of a logarithm from 'm' to 'n'. The result is a rational expression in which the numerator is a logarithm of 'a' with base 'n' and the denominator is the logarithm of 'm' (which was the previous base) with base 'n'. Thus, the resultant expression has all logarithms in the base 'n'.
  • a: argument of the logarithm
  • m: original base
  • n: new base

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