In this case, y(x) = b

^{x}so we want

As he says, the key idea is that we can split the first part

and then factor out b

^{x}and move it outside the limit:

The term in the limit is something. It depends on what the limit converges too. However, the important thing is that it is not a variable but a constant. Thus,

This is the equation that I used in the last post. Starting from here there is a quick demonstration of something that we proved in a more roundabout fashion before. Start with the equation above and invert it:

but

Beautiful! Strang says "that proof was...powerful too quick." So he does the following.

f(y) converts y to x, it takes the logarithm to base b of x.