Welcome ladies and gentlemen.
So what I'd like to do is show you
how to expand logarithmic expressions.
Now expanding logarithmic expressions,
what we're going to do is follow the rules of logarithms.
And the rules of logarithms we have
product rule, quotient rule, as well as power rule.
And I've put them up here, here's the product rule,
here's the quotient rule, here's the power rule.
And then I wrote two other little rules
of exponents that could come in handy as far as rational
powers.
You can rewrite as a radical, and vice versa,
negative powers you can rewrite in the denominator.
So there's multiple different ways to do this.
I'm just going to do it the way that I think would be best.
But just so you know, there's different ways
for some of these problems.
We'll still arrive at the exact same simplified expression
or expanded expression.
So basically what we have when we're expanding
is we're given one single logarithmic expression,
and what we want to do is expand it.
And we're going to basically be using these three properties
of logarithms to do just that.
So basically what we want to do is
first kind of identify what type of property that we have.
So in this case, I have log base 3 of 5x.
Well 5x is multiplication.
Right?
You're multiplying the 5 and the x.
Well what the properties of logarithm state
is if you have the logarithm of a base
b of the product of two terms, what we can do
is rewrite that as two separate logarithms with that same base,
by adding the two logarithms.
So this one is just going to be log base 3 of 5,
plus log base 3 of x.
The next example is going to be using
the quotient property, you can see
that I am dividing my x and y.
So just like I'm dividing my m and the n ,
I'm now going to rewrite that as subtraction.
And again it doesn't matter, I use
the properties of logarithms for base 10,
or for any regular logarithms, but you it
works the same for natural logarithms
as well as a logarithm for any base.
So therefore this is going to be the ln of x minus ln of y.
The last major property is the power rule,
and basically what the power rule of logarithms states
is if you have a logarithm of base b of m,
raise to the n power, we can rewrite that
as the product of that power times your logarithm.
So we can take 4 and rewrite it in front and write log x.
Now, the rules of logarithms are not just-- I just
did one property a time.
But now what we're going to do is start getting into logs
simplifying or expanding logarithms
where there's going to be multiple properties involved.
On this next one it might not be apparent on what exactly
I can do where the radical come from right that's not
available in there well by using our rules of exponents
we understand that a radical can be rewritten as a power.
So I can rewrite this as log base 2 of y to the 1/3.
Now I can take that 1/3 and put it in front.
OK?
In the next one you can see now I
have three different logarithms.
Well with each logarithm I am multiplying.
So now all I'm going to do is apply the product rule,
not once but twice.
I need to separate these two as multiplication, and these two.
Now, you can do it in certain steps, right?
You can do one at a time, or you can just
break up each one of these.
So now I'm going to do log of y squared
plus log of x to the negative 1/3
plus log of the square root of z.
OK?
Now the main important thing here
is you can see that now I have powers in each one,
well not in each one of those.
I can rewrite this as raise to the 1/2.
Right?
Well now you can see each of my logarithms have a power.
Well we don't like having powers, right?
We want to write those powers in front.
So to write it in fully expanded form I'm
going to bring the 2 down, I'm going
to bring down the negative 3, and I'm
going to bring down the 1/2.
And usually plus or minus we just write it as minus.
OK?
But there you go, that would be our fully expanded form.
In the next one here you can see that now I
have the product rule over the product
rule and the quotient rule.
So rather than doing the product and then the power,
now I have two properties.
Now what I would recommend when doing
this is to separate them using the quotient rule first,
and then using the product rule.
So if you were to do that it looks something like this,
log of 2x minus log of y squared.
OK?
So by doing that, I've separated them using the question rule,
now I can apply the product rule here
and the power rule over here.
So I can do log of 2 plus log x minus, bring the 2 down,
2 times log of y.
Over here I have the exact same case except now
I guess I just have some more numbers in the bottom, top.
Actually, you know what, let's bring that-- that's
like the exact same problem.
Let's put the [INAUDIBLE] down below.
Now again, this is going to be the exact same thing,
separate them by the quotient rule, then do the power rule.
So it's going to be ln of y minus ln of x cubed z squared.
OK?
Well that's already done, so I just do ln of y minus,
I'm going to separate them, ln of x cubed plus ln of z
squared, then I apply the power rule.
OK?
So a lot of times we're going to be using multiple properties
in a row.
I think it's just helpful whenever
you have a rational term to separate
using from the quotient rule first
and then apply certain properties.
However, in a problem like this we can't simply
do that, because our rational term is being raised to--
or you're taking the 5th root of it.
So the first thing I would do is rewrite this as a power.
Now again, we can't apply the quotient rule
because our quotient is being raised to the 1/5 power.
So the first thing I'm going to do
is actually use the power rule to get rid of that power.
Now I don't have a power over my x divided by y,
now I can separate using the quotient rule.
And there's a big problem we have to be very careful with,
and I'm going to show it to you here in just a second.
Now if you look at what I just wrote,
I have 1/5 log base 5 of x minus log base 5 of y.
Well this is only saying that the exponent is only
going to go to the x, it's not going to go to the y.
So we have to be able to devise a symbol that would represent
that this 1/5 is really being raised to both of these, right?
So we need to include parentheses or brackets,
but if you don't produce the parentheses or brackets it's
going to be wrong.
So you have to be very, very careful with that.
OK, next example.
Here I have a quotient so I can separate it.
So that's what I'll do first.
I have log base 6 of 25 minus log base 6
of the square root of x plus 1.
Now this one gets a lot of students
because when we're looking at this we say, OK well
I can rewrite the square root as x plus 1.
But what do I do when there's-- what do I do when I have terms
separated by addition?
Nothing.
This is your term, this is your expression, this is like your x
or is like your y.
We can't separate these using the properties of logarithms.
The properties of logarithms only
work for multiplication, division, and raising
to a power.
So this is like our expression.
If you want to think about it just
erase it and think of it just as another variable.
OK?
And just follow the same process.
But now, what we can look at here
is basically all you really need to do
now is put the 1/2 in front.
So my final answer is log base 6 of 25 minus-- oh you know what?
I didn't do that.
I'm going to change this one more time.
I'm going to put that to the third power.
So I have 6 to the 25 equals 1/2 log base 6 to the x plus 1.
So again be very, very careful with keeping that there.
All right, the reason I wanted to put that 3 there
is because it's kind of similar to the same thing
here, we have to make sure we keep those parentheses.
So again, the first thing whenever I have a power I
want to bring that power in front.
I'm going to bring the power in front
and I'm going to apply the quotient rule.
I'm kind of running out of space,
so I'm going to do a couple properties at the same time.
So by applying the quotient rule--
and I'm going to put that x minus 3 in parentheses,
because again, just like here, it's a product all into itself.
And again, I want to make sure I put parentheses
around this because this 3, if you
were to condense these I would need to put the 3 up
to both of them.
So it's really, really important that you
keep those parentheses.
That's why I wanted to add the 3 to this problem.
All right.
Well I can now play the power rule.
So that's going to be ln of x to the 1/2 plus ln of y squared
minus ln of x minus 3.
And now I can bring my powers in front.
And I have now finished expanding
my logarithmic equations.
Thanks.
Or expressions.
Thanks.