There are five properties of logarithms
that you should know. These are the first
three: log base a of X plus log base a of
Y can be rewritten as a single logarithm,
log base a of - you multiply within your
parentheses x times y is XY. So if you
add two logarithms together, you can
multiply what's inside their parentheses
and write it as a single logarithm. Same
idea: log base a of X minus log base a of
Y. This time if you want to write it as a
single logarithm, instead of multiplying,
divide. X divided by y. The negative log
piece goes in the denominator.
Alright, and then last one: it's already written as a single logarithm, but if
you have any kind of coefficient or
number in front of your logarithm like
this -  x times log base a of Y - you can
bring that X inside your parentheses as
an exponent. So this is log base a of Y
to the X power. Whenever you start with
multiple logarithms or with a
coefficient and rewrite it as a single
logarithm or without a coefficient, it's
called condensing. So if you go from the
left side to the right side you are
condensing or writing it as a single
logarithm. Whenever you start here with a
single logarithm and move all of your
exponents out or split it apart into
multiple logs,
it's called expanding. Sidenote: in
order for you to use this property, these
bases here must be the exact same number. In this class that's always going to be
the case so you won't really have to
worry about that rule. Okay let's do I
have five examples of this. Your directions are usually going to be
to "Condense," but you may also see it as
"Express as a single logarithm."
So we'll start basic: log base 2 a 5 plus
log base 2 of 6. This is two logarithms
being added together. I can write it as
one single logarithm,
and because I'm adding here, you multiply
the 5 and the 6 together. And because you
can actually multiply 5 times 6, it's
silly to leave it separated like that. So
5 times 6 is 30. Alright, let's do some
that are not just using one property or
why would you ever use that third
property? The reason you use this third
property that deals with exponents is
because you have to get rid of any
coefficients before you can combine and
use these first two properties here. So
when we're condensing, get rid of your
coefficients first by moving them in as
exponents. Then you can combine your
logarithms.
When condensing, deal with your
coefficients - that is the number in front
of your logs - before writing as one
single logarithm. Okay, so if I bring that
3 inside it's only brought into this X
here. 3 natural log of X minus natural
log of 4. I like to write my natural logs
in cursive because it helps me keep it
straight that it's an L and not an I
you're welcome to write them however you
want. Anyhow, if I bring that 3 inside
it's only going to go with the X. I'm
cubing the X and the 4 stays as is. And
now I can use the second property
here that says when you subtract your
logs you can write it as a single
logarithm and you would divide your
arguments is what you call what's
written in your parentheses. So the minus
sign means the four goes in the
denominator. I'm gonna - well I'll write
that in the next example.
Okay, the minus sign means you put it in
the denominator. Alright, let's do one
that's a little more difficult. Same idea,
though. We have to deal with our
coefficients before we can write it as
one log. So I need to bring this three
inside as an exponent, this nine inside
as an exponent, and this five inside as
an exponent - just the number you don't
have to worry about bringing the sign
inside. So this would be rewritten log base 7 of x cubed plus log
base 7 of Y to the ninth power minus log
base 7 of Z to the fifth power. I can
combine all three of these into one
logarithm in one step. You just have to
remember minus means denominator. So this
piece here has a minus in front of it. So
it is going in the denominator. Everything else stays on top of your
fraction. Log base seven. The X cubed:
there's not a minus sign in front, so X
cubed stays on top. Y to the ninth: there's
not a minus sign in front, so y to the
ninth stays on top. Z to the fifth: there
is a minus sign on top, so it goes in the
denominator. And I skipped an example
from what's in the notes. It doesn't
matter what order you have everything
written in (this could be a positive.
Actually, let's change it from what's in
the notes. This could be a positive), it
just matters that the minus piece: 
this is going to go in the denominator. The
minus piece goes in the denominator. Everything else stays on top. So I just
changed this problem from what is in the
notes, but we can still work through it.
Just like before, deal with your
coefficients first. Nothing there to
worry about. My 1/3 comes in here. My 2
comes in here. So this can be rewritten
log base 5 of X minus log base 5 of Y to
the 1/3 power plus log base 5 of Z
squared. Now, rewrite it as one single
logarithm. Log base 5. Anything with a
plus sign in front of it or anything
that doesn't have a negative (if there's
not a negative it's implied to be a
positive) so any positives go on top. Y to
the 1/3 has a minus in front of the log,
so it goes in the denominator. Z squared
has a plus in front of it because we
changed this problem from our notes, so Z
squared stayed on top. This is how ALEKS, your homework system, wants your answer.
But just as a side note, let me write
that in blue up here. If I have a
fractional exponent like this, usually
you rewrite it as a square root. And this
denominator here becomes your index and
your a the numerator stays as your
exponent. So your homework allows this to be your answer which means I will allow
that on your test as well, but just know
that generally you would rewrite this as
instead of to the 1/3 power, you would
write it as the cubed root. You can have
a 1 there if you want, so this one is
usually considered more correct, but
they're both correct according to your
homework. Let's do one last example. If
you plan to take calculus, you're more
likely to see something like this where
you have binomials in your parenthesis instead of just single numbers
or monomials that just have a mixture of
X's and numbers with no plus or minus.
The rules don't change. We're still going
to deal with your coefficients first and
minus still means move in the denominator. The
only difference here is when you bring
this three inside your parentheses,
you're not really bringing it inside the
parentheses or we need to create new
parentheses because this will be natural
log of X plus 2 ALL cubed. You don't
distribute that cubed to the X and to the
plus two. Same thing here: this squared.
You create new parentheses to hold that
four X minus seven together, and the four
X minus seven as a whole is going to be
squared now. If I wanted to write this as
one logarithm, because they're added
together or they're all plus signs that
means they stay on the top of my
fraction, so I will have X plus two all
cubed and 4 X minus seven all squared.
This looks long and messy, but you will
leave it like that. I have a few extra
that you may want to practice before you
get to your homework. So you may want to
pause the video here, and try these three
problems. Here are the answers. For this
first one, you move your six inside
your parentheses and then, because this
is a minus, it goes in the denominator. On top, both of these are pluses so they stay on
top of the fraction. You would multiply
three times two together to get a six.
For the second one, bring your two inside
your parentheses. You'll have to create
some new parentheses. And bring your
three here as an exponent. 2 cubed is 8.
And then for the bottom or the last one,
bring your squared inside, your 4
inside, and anything that has a minus
sign moves to the denominator. So 5 and 5 X
+ 1 to the fourth power stay on top, and
3 squared which is a 9 is in the
denominator.
