In this example, we're going to
take a logarithmic expression
and express it as a sum or
difference of logarithms.
We'll also express
any powers that we see
inside this expression as factors.
And in doing so, we'll expand
it as much as possible.
So to start, let's review a
few of our logarithm rules.
So the rules that we're
going to use here,
we're going to take any
products-- so anytime
we have a logarithm
of a product, we'll
expand it into the
sum of two logarithms.
And that is Property 5
that we're looking at.
Well, also, anytime we see
the logarithm of a quotient,
we'll expand that into the
difference of two logarithms.
And the third thing we'll do
is relating to that power part
that I was just talking about.
So any time we have an
exponent inside our power,
inside that logarithm, we will bring
that out front as a coefficient.
And that's our
Property 7 here where,
anytime we have a
log base a of some M
raised to the r, that's equivalent
to taking r times log base A of M.
So again, that power r can come out
front as a factor-- only when it's
inside the logarithm, though.
OK.
So let's look at our example.
We have the natural
log of this quantity,
and then we have the
quotient of the square root
of the quantity x squared
plus 4 over the quantity
x squared minus 4.
So we're going to deal with
the natural log function
throughout this entire example.
So let's take that
original expression,
and we have our square
root here of x squared
plus 4 over x squared minus 4.
All right.
Now, the first thing
we want to do, we
have a quotient as that
main operation there.
So we have division as
our main operation there.
So we're going to turn this
then into the natural log
of that numerator, which
is the square root of x
squared plus 4 minus the natural
log of x squared minus 4.
And if you're going back to those
properties that we were looking at,
that's Property Number
6 that we used there.
OK.
So we've got that split
into a difference now.
From here, there's a
couple things that we're
going to want to do algebraically
in order to expand this more.
And the first is that the
square root that we're taking,
we're taking the square root of
the expression x squared plus 4.
And so this can be
written as x squared
plus 4 raised to the 1/2 power.
That's the first thing
we're going to want to do
to apply more logarithm rules.
The second thing is that x
squared minus 4 can be factored.
So we'll turn this into
the natural log of then
x minus 2 times x plus 2.
We'll factor of x squared minus 4.
OK.
So now that we've done this,
we can apply that power rule.
We see that we have this
1/2 in the exponent.
So that will become then
1/2 times the natural log
of x squared plus 4, and the
x squared plus 4 is prime.
So that natural log of x squared
plus 4 is done being simplified.
Looking at the second
logarithmic expression we have,
we have the natural log of
this x minus 2 times x plus 2.
And that's a product.
So we're going to be
able to then split this
into the natural log of that
first factor, which is x minus 2,
plus the natural log of x plus 2.
And if we're looking at
referring to those properties,
then that's going to be Property
Number 7 for that first expression
and Number 5 for that second one.
The last thing we want to
do is expand this further.
So we have-- we're
subtracting then this sum.
So we'll subtract each
term in that sum then.
So this will be 1/2 times the
natural log of x squared plus 4
and then minus the
natural log of x minus 2
and then minus the
natural log of x plus 2.
And at this point, everything is
expanded as much as it can be.
We can't do anything with this
natural log of x squared plus 4.
We can't do anything
with the natural log
of x minus 2 or anything further
with the natural log of x plus 2.
