Welcome to a lesson on how to use
the power rule or power property of logarithms,
which is log base b of M raised to the power of n,
equals n times log base b of M.
On the left notice how n is the exponent
of the number part of the logarithm
and on the right n is the coefficient of the logarithm.
You'll be using this property first to expand logarithms
and then to write logarithms with a coefficient of one.
With the first example we're asked to expand the logarithm
using the power rule or power property of logarithms.
We're given log base seven of x to the fifth,
which can also be written as log base seven
of x to the fifth in parentheses.
Now, we know x to the fifth is equal to five factors of x,
so we could expand this using
the product property of logarithms shown here,
which would give us a sum of five logarithms.
However, we're specifically told to use the power rule,
or power property of logarithms.
So referring back to the prompt I notice how the exponent n
in this logarithm is equal to five,
which means log base seven of x to the fifth
is equal to five times log base seven of x.
Another logarithm is expanded using
the power property of logarithms.
For number two, given log base two of 81,
let us write now the number part of the log
is not written using exponents, so they cannot apply
the power property of logarithms in this form.
So let's determine the prime factorization of 81
so we can write 81 in the form
of M raised to the power of n.
81 is equal to nine times nine,
so we could write 81 as nine squared,
then apply the power property of logarithms.
But instead, let's look at the prime factorization.
Nine is equal to three times three here as well as here.
81 is equal to four factors of three,
which means 81 is equal to three raised
to the power of four or three to the fourth.
So log base two of 81 equals
log base two of three to the fourth.
Whenever the number part of the logarithm
is written in exponential form
we can apply the power property of logarithms.
We always have the exponent n as equal to four,
which means log base two of three to the fourth
is equal to four times log base two of three.
And now we use the property in the reverse order
to write logarithms with a coefficient of one.
And this is important because
notice how in order to apply the product property
and quotient property of logarithms
from the right side to the left side,
the coefficient of the logarithms must be equal to one.
Since for number one we're given three natural log x,
so looking at the power property of logarithms,
notice how the coefficient n is equal to three
which means three natural log x equals
natural log x raised to the power of three or x cubed.
On the left three is the coefficient,
on the right three is the exponent.
And then for number two we're given 1/2 log base three
with a quantity x minus two.
Here the coefficient is 1/2,
which means 1/2 log three with a quantity x minus two
is equal to log base three
with a quantity x minus two raised to the 1/2 power.
The logarithm does have a coefficient of one now,
but having a quantity raised to the 1/2 power
is the same as taking the square root,
so we can also write this as log base three
of the square root of the quantity x minus two.
These two logarithms are equivalent.
One is written using irrational exponent,
and one is written using the square root.
I hope you found this helpful.