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.