Let's look at some other properties of
logarithms. We have three here. We'll go
through each one and then we'll do an
example using them. First one, we have,
call this the product property, log base
a of x times y, so here we have x times y
in the argument of logarithms. We can
write that as a log base a of X plus the
log base a of Y. For division, in the
argument X over Y log base a of X over Y
is log base a of X minus log base a of Y,
and then finally here we have this
so-called power property. I think it's
the most important one. Log base a of X
raised to a power R is R times the log
base a of X. So we can take the power
from the base here, this X, and bring
it down as a coefficient of the log base
a. Alright, one reason we have these
properties and one use of them is to
expand a logarithmic expression. Like
this one. Why would we want to do that?
Well, as you can see the argument of this
logarithm is very complicated. If we
expand it using these properties, we get
a bunch of very simple logarithms at the
end, and that's useful for other
mathematical activities, so let's expand
this. Alright, so we have the natural log
of the square root of x cubed times P
times Q to the fifth power all over e to
the seventh. Now, the first thing we want
to notice about the argument is that it
is a quotient, so it is something over
something else. Even though it's very
complicated there's a lot of terms in
there, it's really just something over,
it's really an x over a Y, right. Some
expression over another expression, and
we can write that then as the log base a
of the numerator minus the log base a of
the denominator. So, we can write the
natural log of square root x cubed times
P times Q to the fifth
minus natural log e to the seventh. Alright,
now we have the natural log of a product,
product of these things, square root of x
cubed P, Q to the fifth. We can use the
first property to split that up into a
bunch of sums of logarithms, so of the
natural log of this. Natural log of the
square root of x cubed plus the natural
log of P plus again, the natural log of Q
to the fifth. Alright, minus, and now we have
the natural log of e to the seventh. We
recall that the natural log and the
exponential function base e are inverses
of each other, so if we take the natural
log of e to the seventh, we're left with
just the exponent, seven. Alright, what
else can we do to this? Well, notice that
we have some powers. We've got a square
root and a third power here on this
argument. We have a five on the
exponent of Q, so we can use this
property to bring down the exponents.
Let's look at the square root of x cubed.
Square root of x cubed can be written as
a fractional exponent like this. We know
that the square root is one-half power,
and then we can evaluate this by taking
the product of these two exponents, so
this is actually X to the three halves.
So what we have right here is the
natural log of X to the three halves. Now
using this property, we can bring down
that exponent as a coefficient. So three
halves
natural log of X, okay, plus natural log
of P, and then here we have a five on the
exponent of Q. We can bring that down as
a coefficient as well. Five natural log
of Q minus
seven. Alright, so our original
logarithm can be expanded as three
halves natural log of X plus the natural
log of P plus five natural log of Q
minus seven.
