In this video we're going
to look at the power rule.
And we're going to use
the power rule to expand
logarithmic expressions.
And this is the power
rule for logarithms.
It basically states
that if m and b
are positive real numbers
and b is not equal to 1
and let r be any real
number that the log base
b of m to the r is equal to
r times the log base b of m.
So the proof is in
a separate video.
And just as before I encourage
you to look at the proof
so that you can
understand why you're
allowed to use this property.
This video is really
intended on showing you
how to use the property.
But on kind of a related rule
is the power rule for exponents.
When you take x to
the m and raise it
to the n it's equal
to x to the m times n.
And so when you take a power
and raise it to a power
you multiply the exponents.
And so in a loose related way
basically a log is a power.
And you're raising m to a power.
And in the end you taking r and
multiplying it by a log, which
is a power, an exponent.
But let's use this property
to expand these expressions.
So we'll write the rule
base b of m to the r.
That's a little wide.
That's equal to r times
the log base b of m.
And so for this
expression right here
the 5 is going to come
down in front as a factor.
5 times the log of 8 base 6.
So generally when
you're asked to expand
a logarithmic expression
you take any exponents
and rewrite them as powers.
And so for the next one, for
b, before we use the power rule
we want to rewrite the square
root of 10 to 10 to the 1/2.
And then we'll multiply the
log of 10 times the 1/2.
And we know that the
log of 10 is equal to 1.
So if we just multiply
1/2 times 1 we'll get 1/2.
On the last one I'm going to
show you two different ways
to do this one.
One is to use the power rule
first and then simplify.
And then the other would be
we'd expand a little bit.
We'd raise that expression
to the power of 3 first
and then use the power rule.
So let's first use
the power rule.
We'll bring up 3 down
as a factor first.
Now I'm multiplying 3
times the log of a product.
And the log of a product
is the sum of the logs.
Now the log of 7a is the
log of 7 plus the log of a.
And since I'm multiplying
the log of a product
I now have to multiply a sum.
And without the
parentheses here I'm
really just multiplying
the log of 7
by 3, when in fact I need to
multiply the whole sum by 3.
Then we'll distribute the 3.
And this is considered expanded.
Your second option
on this problem
would be to raise the 7 and
the a the exponent first.
So we'd have 7 cubed, a cubed.
Then expand.
So now I've got the
log of 7 cubed a cubed.
When you take a product
and raise it to a power
each of those factors
gets the exponent.
I'm going to get the log of 7
cubed plus the log of a cubed.
The 3 comes down.
3 log of 7 plus 3 log a.
Just a word of caution, the
power rule log of x to the r
is not the same as saying
the whole logarithm raised
to the r power.
So you've got to
be careful there.
It does not mean the same thing.
So the question is, what's
the big deal about this rule?
Why would it even
benefit us at all?
So if you think about if you're
working on a problem like this
without a calculator and
you wanted to evaluate this.
Well, 8 cubed if you multiply
that out you get 512.
So I've got the
log base 2 of 512.
And if I want to evaluate
that I'd say well let's
let that equal x.
And then I'd say 2 to
the x is equal to 512.
And then literally I could play
around with trial and error.
I mean without a calculator
this would be a bear.
I'd have to keep multiplying
2 a number of times
until I got to 512.
And the answer is 9.
If I multiply 2 by itself
nine times I will get 512.
But if we use the power rule,
and obviously these properties
and rules were
developed at a time
we didn't have calculators.
So this made a huge
difference in evaluating.
If I bring that 3 down I have
3 times the log of 8 base 2.
And I know I can evaluate the
log of 8 base 2 pretty quickly.
It's 2 to what power gives me 8.
That's going to be 3.
So 3 times 3 equals 9.
So here's an example where
the power rule is extremely
helpful in evaluating.
It's also helpful in calculus
if you can rewrite an expression
rather than have it
in exponential form
to be able to write
it as a product.
So what I'd like you to do
next is pause the video.
And I'd like you to expand some
more logarithmic expressions.
And in some of those
we're including
the power rule in there.
So the first one we
just want to expand.
We're not asking
to evaluate, just
want to expand the expression.
So the 4 comes down as a factor.
And it's the log base 5 of 10.
The next one, before
I expand it I'm
going to write the y to the 1/3.
And now the 1/3 comes
down as a factor.
And it's 1/3 times the log of y.
Here we have the argument is
being raised to the 2nd power.
Remember that's
not the same thing
as the natural log of 3x minus
5 raised to the 2nd power.
In this case, we wouldn't be
able to use the power rule.
But here we can.
It's 2 times the natural
log of 3x minus 5.
And since we're not
allowed to expand a sum
or difference than this is as
far as you can go with this.
This is expanded.
