Proof of Log Properties: Rule 3

00:01 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.

00:11 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.

00:19 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.

00:49 So if we raise this by a power of n, we should raise it by the other side as well, just to be fair.

00:56 So we can say x to the n is a to the p to the power of n.

01:01 So because I'm doing the same thing on both side of the equation, I'm not really altering this equation.

01:07 This we can rewrite as x to the n, equals to a and then these two numbers multiply to the np or pn.

01:15 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.

01:31 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.

01:51 So we proved the third rule of logs.