>> Here we're going to
use the Laws of Logarithms
to expand expressions.
We will want to expand
the expressions
so that there are no
unnecessary products, quotients,
or powers in the argument.
So in the first case
we have log base a of x
to the eighth divided
by y z to the ninth.
So first of all,
this is a quotient.
So we'll use the Quotient
Rule to write log base a of x
to the eighth minus log
base a of y z to the ninth.
Now, we can also use the
Product Rule on the second term.
So I'm going to leave the
first term alone for a moment,
and I'm going to use
the Product Rule.
Be careful, we will need a set
of parentheses around the sum.
So we'll have log base a
of y plus log base
a of z to the ninth.
We have to subtract that sum,
so that's going to
change our signs.
We will, remove
the parentheses..
So minus log base a of y minus
log base a of z to the ninth.
And lastly, we'll use the
Power Rule, which says instead
of having an exponent in the
argument that exponent can come
out front as a coefficient.
So we'll have 8 log base
a of x minus log base a
of y minus 9 log base a of
z.... In the second example,
we have log base 2
of the square root
of the quotient, x minus 1, divided
by , x plus 1.
So first of all, we'll write
the radical as the power..
x minus 1, divided by, x
plus 1, to the 1/2 power.
And then bring 1/2 down as
the coefficient of the term;
1/2 log base 2 of the quotient
x minus 1, divided by, x plus 1.
Now, we will be able to
use the Quotient Rule.
But, again, be careful
with parentheses.
And it will be log base 2
of x minus 1, minus, log
base 2 of x plus 1.
The preferred form of this
would be distributing the 1/2
to get 1/2 log base 2
of the quantity, x minus
1, minus 1/2 log base 2
of the quantity, x plus 1.
Our last example here is a
natural log of a quotient.
So we have.. we'll have the
difference of natural logs.
And we'll have natural
log of the numerator..
which is 7 x to the eighth.. minus
natural log of the denominator,
which is x plus 9, squared.
Now, in the first term, we're..
we can use the Product Rule.
And so this would be the natural
log of 7 plus the natural log
of x to the eighth, minus, the
natural log of, x plus 9, squared.
And now we have exponents
of 8 and 2
that can be brought
down as coefficients.
We'll have the natural
log of 7, plus 8,
natural log of x, minus 2
natural log of the
quantity x plus 9.
