Welcome, ladies and gentlemen.
So what I'd like
to do is show you
how to condense
logarithmic expressions.
So what I did is
I wrote up here.
I wrote some up top-- I
wrote the common property
of logarithms to
help us condense.
And what I did is I wrote the
expanded form of a logarithm.
And then I wrote what the
equivalent expression for that
is in the condensed format.
So you can see I have one
example here, another logarithm
here, another example there.
So I kind of ran out a little
bit, got tight on space.
But I put each logarithm.
So this would be basically
your product property,
your quotient property, and here
it'd be basically your power
property.
And in another video, I explain
how those kind of relate.
And they're similar to the
rules of exponents and so forth.
But the main important thing,
the really kind of idea,
and we're going to use all
of these properties here--
and what we're
trying to do is we're
trying to condense our
logarithmic expression.
You can see that these
expressions are separated
by addition and subtraction.
And what we're going
to do is we want
to condense these into one
single logarithmic expression.
So to do that, first of
all, if you can see here,
by using my addition
property, when
I have two logarithms
separated by addition,
I can condense them into the
product of our two values
of the logarithms, x and y.
So in this case, I have log base
4 of 20 plus log base 4 of 3.
Well, therefore, that's going to
equal log base 4 of 20 times 3.
Now, for some of these, I'll
break them out step by step.
And some of these
I'm going to run out
of a little bit of
a space, as well.
However, that can simplify
now to the log base 4 of 60.
Now, we always want to
see if we can simplify
this a little bit further.
Some of these I chose, we
will be able to simplify.
Some you will not.
We know that 4 raised to
the third power is 64.
But I can't really simplify
this any other further
to be able to evaluate.
So I'm just going to leave
it as log base 4 of 60.
In the next problem here, now
you can see we have division.
So to do this next
problem, when you
have terms of logarithms
separated by addition,
you can rewrite
them as the product.
So when you have terms
separated by subtraction,
we can rewrite them
as the quotient
of one single logarithm.
So this would be log base
2 of 18 divided by 3.
Well, 18 divided by
3 is going to be 6.
So therefore, I can rewrite
this as log base 2 of 6.
And again, that's
not something I
can simplify because 2
doesn't raise to a given power
to give me 6.
So therefore, I'm just going
to leave my answer as is.
So hopefully, ladies
and gentlemen,
you just kind of get an
idea of what we're doing.
The main important thing
is taking these expressions
and condensing them down to
one logarithmic expression.
So now I'm going to start
incorporating another rule.
And that other rule
is the power rule.
And what the power
rule basically states
is if you have a value
that's being multiplied
by your logarithm,
in front of it,
you can basically rewrite
that as your power
of your logarithm.
So in this case,
what I'm going to do
is I'm going to apply that power
rule for each of my powers.
So therefore, I have log of
x squared plus log of y cubed
plus log of z to the fourth.
Now, also notice that
the logarithms do not
have a base in this case.
My logarithms does
not have a case.
But fear not.
When your logarithms does
not have a base-- oh,
I forgot something.
That's supposed to be x plus.
This is supposed to be 2x minus.
Let's do it like that.
There you go.
So when they don't
have a base, just
remember if your logarithm
doesn't have a base,
then we can assume it's base 10.
Now, that's not really going
to affect our condensing.
But it is important we can
only apply these rules when
the base is exactly the same.
So I didn't do any examples
here because there's really not
much to do for it.
But if you have a
logarithmic expression
that you're trying to
condense and the bases are
different-- for instance, if I
had like 5 and 10 or something,
or 5 and 2-- you couldn't
follow these rules
of order of operations.
So if there's not
a base given, we
assume that it's going to be 10.
Well, now I have addition
and I have addition.
So I can rewrite this
as one single logarithm,
as the product of
my three values.
So therefore, I just
write this as log of-- I
always write this.
I don't need parenthesis--
x squared, y cubed,
z to the fourth.
Now, I forgot to
write this one in.
But in the next example, now
we have natural logarithm, ln.
And natural logarithm
is the same thing.
I don't have any room here.
Well, I guess I can write here.
So log base a of x is the
same thing as ln of x.
And basically what
that means is ln
of x just tells us that
it's going to have a base e.
So again, this is going to
be the exact same thing.
Nothing's really changing.
It's just telling us our
base is now going to be e.
Well, we're going to
apply the same operations.
So in this case, though,
if I'm applying-- now
I'm applying this
[INAUDIBLE] quotient to 2.
So you've got to
be really careful.
You're not just applying
the power or the x.
So I'm going to bring that up.
And then here, I
don't have a value.
So now let's rewrite
this as ln of 3-- sorry.
So that's going to
be 3x squared, which
is now going to give
me 9x plus ln of 4x,
and then minus ln of 2x.
Now, what gets everybody
confused here is they want to,
again, use this quotient rule.
Well, we can't separate
the 2x minus the 1.
So that is an expression
all on itself.
So since that's
within parenthesis,
that's going to remain there.
That's what actually we're
going to dividing our value by.
Now, in the next cases,
when you're condensing,
we basically want to follow
our rules of operations.
Do the product rule.
Do the quotient rule.
We always want to work
from left to right.
Unless we see some
parentheses, we're
always going to work
from left to right.
So my first expression here
is going to be multiplying.
So I'm going to
basically have 9x times
4x, which is going to be 36x.
So I have ln of 36x
minus ln of 2x minus 1.
Well, now I have the
quotient property.
So I'm just going to
rewrite this as one
by dividing 2x minus 1 from 36x.
So my final answer is ln of
36x all divided by 2x minus 1.
And again, that's
going to be a base e.
But that's going to be your
final, simplified answer there.
So let's go over to this one.
I guess I didn't leave myself
some space here, did I?
Well, I guess we'll try
to get through this here
as quickly as possible.
Hopefully, I can.
So in this case,
again, well, let's just
try to simplify this
as much as possible.
So the first thing I
want you to understand
is, again, we can
bring up our powers.
So let's bring up our powers.
And then let's divide here.
So therefore, this can
be written as log base 2
of x cubed divided by y.
And that's going to be plus log
base 2 of z to the 1/2 power.
So I applied the
quotient property
and I bought the 3 up there.
So I'm just doing it
in a couple steps.
Now, we can combine
these by multiplication.
But it's very important
for you to understand
that z to the 1/2 power
is, again, the same thing
as using a radical.
So don't forget our
powers of radicals here.
Here's just another
order of operation.
Remember, if you have x to the m
over n, that's equal to the nth
root of x to the m.
So that's another
property we just
don't want to forget about.
So therefore, now I'm going
to be multiplying this.
Now remember, if
you're multiplying
this times this fraction,
just like if you're
multiplying 3 times 4 over
1, you're going the 3 times
the 4, not times both
of them or not times 1
because that 3 is
technically in the numerator.
So therefore, the
z to the 1/2 is
going to be in the numerator.
So it's going to be
multiplied by this x cubed.
However, my final
simplified condensed form
is going to be log base 2 of
x cubed the square root of z
divided by y.
And that would be my
final answer there.
Over in here, you might want to
replace this and then rewrite
this as the cube root.
So in reality, I'll
rewrite this as log base 5
of 4 plus log base
5 of x to the 2/3.
It might be helpful then to
realize that's log base 5 of 4
plus log base the cube
root of x squared.
Now you can see that
your multiplying them.
So you're going to multiply
4 times that to give you
your final logarithmic
expression, which
is log base 5 of 4 times
the cube root of x squared.
So that's kind of how
those rational powers--
we want to simplify
them, use them
as the cube roots
or square roots,
for instance, because
remember, that's just going
to be a square up in there.
Now, in this case, you can
see we have parenthesis.
So whenever you see
parentheses, make
sure you use the
parenthesis first.
That's just really, really
helpful and important
because-- so it's different
because as you work from left
to right, what you can
see is the problem's going
to be totally different as
far as what you multiply
and what you add.
If you look at these terms, how
you're multiplying and dividing
things together really
matters going left to right.
It really does
make a difference.
So if you're giving
your parentheses,
because usually we
would multiply these
and then divide by this.
But since the parenthesis tells
us to divide these two first,
that's when we're
going want to simplify.
So therefore, I will
rewrite this, though.
That's going to be 3 squared.
So that's technically
going to be a 9.
So to save-- to kind
of do a step here,
that's going to be 3 squared.
Actually, you know what?
I'll have time here
because this is
going to be log base 4 of x to
the seventh minus log base 4
of y to the fourth.
Now we need to divide those.
So therefore, this is
going to be log base 4 of 9
plus log base 4 of
division of these
is going to be x to the
seventh over y to the fourth.
Then to multiply these--
again, that 9 is over 1.
So as I multiply
these, it's giving me
9 times x to the seventh
and 1 times y to the fourth.
So my final answer
is log base 4 of 9x
to the seventh divided
by y to the fourth.
And there you go.
The last one and
one of my favorites
is the division
and the division.
So this, again, gets
you a lot of practice.
Again, work from left to right.
But the first step, I think, is
always just to let's go ahead
and rewrite them as powers.
So I have ln of 2 cubed
minus ln of x cubed
minus ln of x to the fourth.
Now, working from left to right,
I'm going to divide these two.
Well, 2 cubed is
8, so therefore, 8
divided by x cubed.
So therefore, this is going to
be ln of 8 divided by x cubed.
And that's minus ln
of x to the fourth.
Now, technically
here, we're going
to have to divide
this once again.
So therefore, I have ln
of 8 divided by x cubed
divided by x to the fourth.
Now, this one gets
a lot of students,
because they're
like, how do I do?
What do I do here now?
Well, the main
important thing here
is when you have
a fraction divided
by a fraction-- for
instance, if I had 1/3
divided by 1/4, what do you do?
The way that I always
teach my students
is multiply by the reciprocal
on the top and the bottom.
Therefore, any number multiplied
by a reciprocal multiplies to 1
and you're left with 4 over 3.
So in this case, I can basically
put my x to the fourth over 1
and then multiply
by the reciprocal
on the top and the bottom.
And by multiplying by
1 over x to the fourth
on both of these-- hello-- I now
obtain my final answer, which
is ln of 8 divided by x.
Now, remember, when
you're multiplying these,
you've got to add the powers.
So that's going to be
x to the seventh power.
So there you go,
ladies and gentlemen.
That is how you use
the rules of exponents
to condense a
logarithmic expression.
Thanks.
