We want to expand each logarithm as much as possible using our properties of logarithms
given here below. Notice both of our logarithms contain square roots, so the first type will
be to write everything in exponent form. Remember, a square root has an index of two,
so we can write this as the common log of X to the power of one-half. We could write this as the
square root of Y to the power of one-half over the square root of Z to the power of one-half.
And again, since the index is two, we can write this as the common log of X to the
power of one-half times Y to the power of one-half divided by Z to the power of one-half.
We could actually multiply these together if we wanted to. Remember,
X to the power of one-half is the same as X to the power of one-half over one,
so we could write this as the common log of X to the power of one-half plus the common log of Y
to the power of one-half minus the common log of Z to the power of one-half.
Notice how we have a product in the numerator, and we also have this fraction, which is a quotient.
Let's start by expanding this by dealing with this quotient. If we have the log of X divided by Y,
we can write this as a difference of two logarithms. So this is equal to the common
log of the numerator, which is X to the power of one-half plus the common log
of Y to the power of one-half minus the common log of Z to the power of one-half.
Now we can expand this first logarithm because we have a product here, so this will expand into the
sum of two logs. This is equal to the log of X to the power of one-half plus the log of Y
to the power of one-half minus the log of Z to the power of one-half. Then for the last step,
we can apply this power property of logarithms, so we can take the exponents and move them to
the coefficients. So this is equal to one-half log X plus one-half log
Y minus one-half log Z. This is considered the expanded form of the given logarithm.
Now for our second example, the first step will be to rewrite this as the natural
log of X to the power of one-half. Then we'll have the quantity X minus three to
the one-half power because of the square root of the quantity X minus three, and
our denominator is the quantity X plus four. Now because we have a quotient here,
we would have the natural log of the numerator minus the natural log of the denominator.
So we have the natural log of X to the power of one-half times the quantity X minus three to the
one-half minus the natural log of the denominator, which is the quantity X plus four. Now it may be
tempting to expand this into a sum of two logs, but we cannot do that because it would only expand
to a sum if we had a product here, and this is a quotient, so we cannot apply this first rule here.
But since we do have a product here, we can expand this first logarithm into a sum of two
logs. This is equal to the natural log of X to the power of one-half plus the natural log of
the quantity X minus three to the one-half minus the natural log of the quantity X plus four. Now
our last step is going to be to expand this using the power property of logarithms, so we can take
our exponent here and move it to the front, and this exponent here and move it to the front.
So we would have one-half natural log X plus one-half natural log of the quantity X minus
three minus natural log of the quantity X plus four. And again, I know it's tempting to try to
expand this logarithm into a difference and sum, but we can't do that because we can only expand
into a sum or difference if we have a product or a quotient of the number part of the logarithms.
So this is as far as we can go, and this would be considered the expanded form of the given
logarithm. Okay, we'll take a look at some more examples in the next video.