Change of Base
The change of base rule is also the calculator formula we introduced
earlier. We start by writing x as an expoenetial in two different
bases. To compare them we need to write one of the bases in terms
of the other. This is how it goes.

Definition using two different
bases 

Definition. Writing one base in
terms of the other. 

Substitution 

Third law of indices 

Equating exponents. 

Dividing both sides by . 
All Logarithms are much the same
The formula in the second last line is interesting by itself.
If we are working in two particular bases a and b then we can treat
as a constant. This equation then tells us that
for any value of x the logarithm with one base is the same
as the logarithm with the other base except for that constant factor.
The graphs of their functions will also look much the same. This
constant will be a scale factor turning one graph into the other.
