Hello, I am Mr. Tarrou. Ok, let's finish this
lesson about properties of logarithms. We
just did a bunch of examples about expanding
logarithms. We are going to do one more of
that and then a few examples of condensing
logarithms again using the three properties
of logarithms we have right here. Ok, so let's
start off with log base four of the square
root of x times y squared over seventeen.
Ok. You do not want to work with logarithms,
excuse me, you don't want to work with radicals.
They are nice for final answers and you are
comfortable maybe at this point in mathematics.
But, once you get used to fractional exponents
where it is power over root, those are much
easier to deal with. So, what we are going
to do is we are going to kind of just go against
the advice I gave you in the previous video.
Normally when you are expanding logarithms
this is the last property you want to deal
with, the power property where you move those
exponents out front. But, since this expression
is in inclosed with on entire large square
root symbol...radical...that means that this
entire fraction that we are trying to expand
is included in this square root symbol. There
is a grouping symbol and all of this expression
is going to... We are going to rewrite it
as log base four of x times y squared over
seventeen all raised to the one-half power.
That one-half power is representing the square
root symbol that is over this fraction. When
you have fractional exponents, the top number
is the power. Now inside this radical we have
exponents of one and exponents of two, and
then we have a constant that has degrees of
zero. But, the entire expression has a power,
or at least the entire fraction underneath
the radical, has a power of one. That is where
this one is coming from and the root is the
bottom number and this was the second root.
So the square root of x y squared over 17
becomes x y squared over 17 to the one-half
power. Now again, we are tying to expand the
log of this fraction which is completely engulfed
by the power of one-half. We are going to
need to deal with that power property first
and move the one-half out front. That is going
to give us 1/2 log base four of x y square
over 17. Now we can start pulling apart this
fraction or expand this fraction. We are going
to first, like in the last video, deal with
this division sign first. So, we are going
to do one-half times log base four of x y
squared minus (that is for the division symbol)
log base four of seventeen. Now why did I
wrap this inside parenthesis and put the one-half
out front? Because it is one-half of this
entire expression which right now is just
one term, but as we expand it to more terms
that one-half is going to need to be distributed
to all of those terms so you do need these
parenthesis and the one-half out front so
that when we are done we can distribute that
through the entire problem. Ok, I want to
pull this multiplication apart so I am going
to have two logarithms added together. One-half
log base four of x plus, for the multiplication,
log base four of y squared minus log base
four of seventeen. Now we are going to do
one last step which is we are going to pull
that two out front using the power property.
Then, we are going to distribute the one-half
through and we are going to be done. We have
one-half times log base four of x plus two
times log base four of y minus log base four
of seventeen. You last step will be to distribute
that one-half and you will be done. One-half
log base four of x plus log base four of y
minus one-half log base four of seventeen.
BAM! Last expansion problem is done. Let's
start doing some condensing problems. Ok...So,
our first condensing problems. The first where
we have two logs and we are trying to write
it as a one singular log. The natural log
of x plus the natural log of five. That is
pretty straight forward. I have not, I have
I written a natural log yet? Yes I have. Ok,
so natural log in the last video anyway I
wrote it... Natural log is log base e. So
natural log of x plus natural log of 5. These
are two logs, both have a leading coefficient
of one, both have a base of e, so we are going
to just put those together and write the natural
log of x times five, or that is kind of unusual,
how about 5x. Ok, one step and BAM! We are
done and moving on. Ok. Wake myself up this
morning. How about one-third log of x plus
log of y? Now these are not going to be just
slapped together like the last example because
we have a coefficient that is not just one.
If you look at our properties, it is one log
base b of m plus one log base b of n. Same
thing with the division rule. You have to
have coefficients of one. So, unlike with
expansion of logarithms which with normally
you want to do our power property last, except
for that last example I did where the whole
fraction was inside the square root. With
compression or when you are condensing logarithmic
functions, a lot of times you do the property
first because you need leading coefficients
again to be one. This is going to come up
and we have the log base ten, log of x to
the one-third power plus log of y. Two logs
with like bases and coefficients of one, and
we get log of x to the 1/3 power times y writing
it as a single log. Our condensing is done.
If you would like to write that as the cube
root of x at this point, that would be ok.
But, that is perfect. No one is going to have
a problem with a fraction exponent since you
have probably been learning about those recently.
Ok, moving on. Let's take a look at five times
the natural log of x plus seven minus seven
times the natural log of x. Now this time
it is subtraction and not addition. So, we
are not going to set up a multiplication,
we are going to set up a quotient or division.
But, like the last problem we have coefficients
which are not one so those come up as powers.
We get the natural log of x plus seven all
raised to the fifth power minus the natural
log of x to the seventh power. Now look, now
we have like bases, we have coefficients of
one, we can go ahead and apply that subtraction
by condensing those logarithms and we will
get the natural log of x plus seven to the
fifth power all divided by x to the seventh
power. Now remember, when you take a log of
a number, natural log...log base ten...log
base of who cares...you do get an exponent.
So this is effectively like the log of something
is going to give you an exponent, the log
of something gives you an exponent (remember
like the log base of ten of a hundred is two,
that is exponent of 2. Ten squared equals
a hundred if you want to undo the log.)...
When do you subtract exponents? You subtract
exponents when you divide with like bases,
and our like base for natural logs is again
e which is approximately 2.718. One more example.
I am not sure how much room this example is
going to take so I am going to erase my properties
here. We are starting off with 1/3 times the
log base two of s plus log base two of y minus
two times the log base two of x plus one.
Ok, well. With logarithms you do want to kind
of want to follow the order of operations...not
kind of...you always want to follow the order
of operations. When you have something inside
of parenthesis that you can clean up, you
know "Please Excuse My Dear Aunt Sally"...parenthesis,
exponents, multiplication and division what
ever comes first as you work left to right,
and then addition and subtraction left to
right, you want to keep those rules in your
head just like they have been hopefully your
whole life. So, we are going to work inside
this parenthesis where we have two logs with
the same base that we can put together first.
This is going to be one-third times the additions
of logs giving us log base two of x times
y, minus two times log base two of x plus
one. Well, this is going to be a little bit
shorter than I thought. Now we are going to
take these two powers or the two coefficients
and move them up to become powers. We are
going to have log base two of xy... Now how
am I going to write the one-third? Because
it is one-third of this entire log, when I
move the one third up it is going to be wrapped
around what I am logging. The 1/3 is not just
on the x, not just on the y, but xy in parenthesis
with the 1/3 power. This all minus log base
two, this is also going to come up as a power
and we already had the parenthesis so this
is probably not a place where my students
would make a mistake, of x plus one squared.
Now that we have just two logs with the same
base and coefficients of one, we can put those
together and remember that we are subtracting
two logs which is effectively like subtracting
exponents...you subtract exponents when you
divide like bases. So, this is going to be
log base two, that common base of the division
that I just mentioned, of you can write (xy)
to the one-third power or you can write the
cube root of xy because remember power over
root...so that is the third root...over x
plus one squared. BAM!!!
