>> We're on to our last logarithm
rule we're going to be looking at.
We've got the power rule, which is very, very
useful for solving exponential equations.
Using logarithms or the natural logarithm law,
we are going to show this one law
is very powerful, very useful.
I'm kind of doing a general base.
It doesn't matter if base 10, base
3, base E, our natural logarithm.
It all works the same.
So let's say I have X to the power of Y.
Okay. I'm starting off very general.
This -- you can do a nice little trick.
Rather than actually having to do X to
the Y, you can rewrite this as Y times log
to the base b of X. There is your power rule.
Later on, we're going to see this is
useful, especially if you have an exponent
where the variable is unknown or that the power
is unknown is X, and you're trying to solve
for X. This is what you are going to be using.
This is a useful trick.
So we're going to look at some of those
applications later when we're solving equations,
but I want to also show you
just applications of this.
So let's go, say, log to the base 2.
And what we might have is -- I'm going to do
a really nice example of this: 2 to the 8.
Hmm. So I can multiply this out.
Get a huge, awkward, ugly number, and then
figure out 2 to the power of what is this?
Well, hopefully you are starting to see
this is going to simplify very easily
because I can rewrite this as 8 log 2 to
the 2 and, sure enough, well, 2 to the 2 --
log 2 to the 2, what does 2
have to be raised to to get 2?
Is 1. So that whole thing becomes 1.
8 times 1 is 8.
Of course that was kind of a silly example
because also if you look at this and go, well,
what does 2 have to be raised
to to become 2 to the 8?
Well, by definition it should be 8.
2 has to be raised to 8 to become 2 to the 8.
But, let's look at another maybe less silly
example, but still demonstrates the point.
Let's go log to the base 5
and say 25 to the power of 3.
This one is maybe not quite
as obvious right away.
Well, we can apply the same rule.
If we wanted to, we could do 25 to the power
of 3, get some huge awkward, ugly number
and then spend a bunch of time playing
on our calculator figuring out 5
to the power of what equals that number.
And, well, we'd have to write down the
number and it would just be tedious.
And why do that when there is
this nice power rule to help us?
We can just bring the 3 in
front -- 3 log 5 of 25.
And now we just have to think, 5
to the power of what gives me 25?
Well, 5 squared.
So this whole thing write here is equal to 2.
This log to the base 5 of 25 just
becomes 2, and so our answer is 6.
And sure enough, try it out
for yourself if you want.
You can go in and put 25 to the power of 3, get
the big huge number, and then do 5 to the power
of 6 and see that it is the
same big, huge number.
So, just quick applications of power law,
and we're going to be using this quite a bit.
So, see you later.
