## The Change of Base Rule

The change of base rule is one of the laws of logarithms and allows us to change the base of a logarithm. It states

We can prove this formula for the change of base rule stating from the equation(1)

Raisingto the poswer of both sides gives

Now take log base b of both sides to give

Finally use (1) to give

Notice that ifthen this becomes

The change of base rule is sometimes useful to solve log equations with more than one log when the bases of the logs are not the same.

Example: Solve the equation

so this equation becomes

Substituteto obtainand rarrange to give

We can factorise this expression to giveand solve it to give

If

If