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.

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.