And now let's look at the last proof, which is the power rule,
which states that log base ax to the n, is the same as n log base a of x.
It's a fairly useful rule this one because it really helps you to bring the power to define,
if you're ever solving index, indices.
So this a little bit easier than the ones that we've done before
because you don't have to make two substitutions,
you can just say let p equals to log base a of x, you can rewrite this in index form,
as a to the p equals to x, then you move to this side, you're starting to prove this part here,
so let's start from the inside x is equal to a to the p,
so we're pretty much just writing the same statement again,
you can then see that this x is raise by a power of n.
So if we raise this by a power of n, we should raise it by the other side as well, just to be fair.
So we can say x to the n is a to the p to the power of n.
So because I'm doing the same thing on both side of the equation, I'm not really altering this equation.
This we can rewrite as x to the n, equals to a and then these two numbers multiply to the np or pn.
And then if we extend this a little bit further, we now have to log both sides,
so we can now say log base a of a and then we have log base a of a,
you can bring np down and then we have log base a x to the power of n.
We know that this equals to one, so that just disappears
and lastly if we use n as it is, and we go back to what we said p was,
so p is log base a of x, so if we write at all neatly log base a, x to the n, is n log base a of x.
So we proved the third rule of logs.