We're asked to expand
the following expression
using properties of logarithms.
We're given log base three of
the product of two X and X minus three
divided by the product of
the quantity two X minus one
and X plus four.
Notice how the argument of the logarithm
is already factored completely
and nothing simplifies.
To begin expanding, we will
first use the quotient property
of logarithms shown here
where log base B of M
divided by N equals log base B of M
minus log base B of N.
Notice how we have the
log of the numerator
minus the log of the denominator.
So in our case, this is
equal to log base three
of the numerator, which is two X
times the quantity X minus three.
Then we have minus log base three
of the denominator, which
is the quantity two X
minus one times the quantity X plus four.
This should be a closed parenthesis here.
Notice now we have a
log of our product here
as well as here,
we can expand further by
using the product property
of logarithms, which is shown here.
Log base B of M times N
equals log base B of M
plus log base B of N.
So for this first logarithm,
we have log base three
of our product of three factors.
One factor is two, one factor is X,
one factor is X minus three.
This will expand into
the sum of three logs.
So next we have log base three of two
plus log base three of X.
Then finally, plus log base three of
the quantity X minus three.
Notice this is also a log of a product.
We need to be careful here
because of the subtraction.
If we subtract this log,
we must also subtract the
expansion of this log.
So because we have a
product of two factors,
this will expand to the sum of two logs,
but we must subtract that sum.
So we'll have minus, then in parentheses
log base three of the first
factor of two X minus one
plus log base three of the second factor
of X plus four.
But for the last step, let's
clear these parentheses
and because of the minus, if it's helpful,
we can think of
distributing a negative one.
So the final expansion
is log base three of two
plus log base three of X
plus log base three of
the quantity X minus three
and then clearing the parentheses
we have minus log base
three of the quantity
two X minus one, minus log base three
of the quantity X plus four.
I hope you found this helpful.
