>> All right.
So Logarithms with
Sum and Difference.
Don't really worry too
much about the formulas,
it's the same rules
we had for exponents.
So if we multiply, we add
the exponents, if we divide,
we subtract the exponents
and if you have a power,
then you multiply them.
So, that's what we're
going to do here.
And I'm going to show
you a nice quick trick
to make them even quicker.
So what you do is you look
at all the different
factors inside this problem,
the divisor and the numerator.
And there are three of them.
So you're going to write
out three logarithms,
one for each of them.
And then you're going to place
each one in that logarithm.
And if it's on the numerator,
that means it's positive,
so on the denominator
it's negative.
If you look at our formulas
that's all that's happening.
See how these are multiplied?
We add them.
They're multiplied, we add them.
This is divided, we subtract
the divisor and that's it.
So if you have log base B of
these, you have two of them,
one on top, one on the bottom,
it inverts the factors,
not the sums.
So log base B again and we have
a Y minus 5 and a 2Y plus 5.
So this one's on top, it's plus.
This one's on the
bottom, it's minus.
That's it.
So what if there's exponents?
Same thing.
There's 1, 2, 3 and then 4,
so we need four logarithms,
one for each of them.
We place each one in there and
their top three are all plus
and the bottom ones a minus.
Now here's the next
step, it's this rule.
If you have an exponent here,
you have to bring it in front.
And is said I explain this
later and here's the reason why.
If you have log of X squared,
let's make it even
better, X to the fourth.
Then rewriting this would be
log, there's four them, log,
log and log, each of them
receiving an X with a plus.
And now, so how many are there?
There are four of them.
So there it is.
Since you have the
4 here, you're going
to get four additions.
So you have to put it
in front as a multiple
so that's why we get to take
that exponent and
put it in front.
So if you look at it this
way, you have two Xs,
so it'd be log X, log X. So
that means you have two of them.
And this one doesn't have an
exponent and this one does.
So that's why you just take the
exponent and put it in front.
And why am I waiting on this?
Because these are both the same
base and anytime you see that,
you have to answer the
question that the log is asking,
which is what's the exponent.
So for a base of 5, what's the
exponent that will give you 5?
It's 1, that one
could be simplified
so we have to simplify it.
And so going back, all
three of these get pluses
because they're on top.
That one is on the
bottom so it gets minus
and then you put the
exponent in front.
So what if we have this?
This is log base 11 and what
we need is to separate them.
We have one giant half, so
we're going to take that half
and give it to every
single one of them.
So it's R to the 3/2, 5 to
the 1/2 and Z to the 2/2.
And 2/2 is 1.
Now we have them all separate,
they all have their exponents
so we need three of them, one
for the R, one for the 5 and one
for the Z. So log 11,
log 11 and log 11.
One for the R, one for the 5,
and one for the Z. I'm not going
to put the exponent on there.
And that one is positive
it's on top,
minus and minus for the bottom.
And then we put the exponent
in front and there you go.
That's our answer.
So all we're doing is expanding
logarithms, returning the logs
into the sum and
difference of logs.
Now we're going to go backwards.
So here's your sum
and difference
and now we have to
put them back.
So we're just going to
work in reverse order.
We're going to put the exponents
back in and then we're going
to squeeze them back
into one logarithm
but as a minus we put it on the
denominator, if it's plus we put
on the top, on the numerator.
So, we're going to
squeeze these back in.
So we have log of 3, log of X
to the fifth minus
log of Y squared.
Now we're going to squeeze
them together as one logarithm.
So positive it goes on top,
positive it goes on top,
minus it goes on the bottom.
And there we go.
So now let's try one more.
So they don't have any exponents
so we can go strait
to the last step.
Plus it goes on top, minus
it goes on the bottom,
plus it goes on top
and that's it.
It doesn't even matter
of the order.
Minus means it's on the bottom
and it's all about
our exponents.
So remember if we bring that in,
it's giving it a
negative exponent
which takes it to
the denominator.
See all the rules still apply.
OK. Give that one a shot.
I need one logarithm
with some stuff in it.
So try it, hit pause
and give it a shot.
So you should have
that and that and that.
So putting it all
together, top, bottom, top.
And there we go, one logarithm
with some stuff in it.
So, if you're given a sum or
difference, you can combine them
into one log when
they ask you to it.
And if you have one log,
you could expand it.
And that's sum and
difference of logarithms.
