Welcome ladies and gentlemen.
So what I'd like to do is show you
how to solve an exponential equation
using the one to one property.
Now what's special about the one to one property
is majority of the times, we don't have to use a calculator.
All of these problems I'm going to solve,
all these exponential equations I'm going to solve,
I'm not going to use a calculator.
Now we do have to have certain equations,
because the next video that I'm going to make
is solving them with using a calculator.
So you can use the one to one property
for every single problem.
But for all of these we can.
And again to kind of remind you of the one to one property,
it's basically just saying if you
had a to the x equals a to the y, then x equals y.
Now that's your kind of formal definition.
And the way I just kind of think about it-- there's
my calculator-- is to think about it in real numbers.
If I had 3 squared is equal to 3 to the x, then
what does x have to equal?
If both sides are equal to each other
and we know one power's 2, the other power's
x, what does x have to be?
Well, it's obvious 2 has to equal x or x has to equal 2.
So that's basically the one to one property.
And a lot of times, sometimes we'll just say they cancel out
and delete them.
But what it means is when you have
the bases are exactly the same, then their powers
are equal to each other.
So what happens with this is I didn't do any basic ones, which
I guess I probably could have.
But you look at 4 to the x equals 16.
Well in this case, we don't have bases
that are exactly the same.
So therefore what I'm going to need to do in this case
is I'm going to need to rewrite this expression with the bases
being exactly the same.
And the reason why the one to one property
works for all of these is because we can rewrite these
all with having the same base.
So you gotta look at 16 and say, can I rewrite 16
as 4 raised to a power?
Well this one should be fairly obvious.
We have 4 to the x.
And I could write 4 squared as equal to 16.
So if I have 4 to the x equals 4 squared,
well now I can just say that x is equal to 2.
And there I go, done.
Now in this next example, we know
that 4 square is equal to 16.
But we have a problem.
This 16 is in the denominator.
So here I have an exponent.
And here I have a fraction.
So one of the things that we need to do
is learn to rewrite fractions with negative powers.
And that comes into one of our rules from negative exponents.
I'll write the one to one property here as well.
So I can rewrite 16 with a negative power.
4 to the x equals 16 to the negative first power.
And again, that comes from our rules of using negative x.
Now I know that 16 I can rewrite as 4 squared.
Just make sure that you're using your parentheses correctly.
Now you're going to want to use the power rule, which remember,
the power rule is when you have an exponent raised
to another power.
You multiply the powers.
So now I'm going to multiply by 2 times negative 1.
So 4 to the x equals 4 times a negative 2.
Now I can say that x is equal to negative 2.
So there's a difference when x equals
2 or x equals negative 2.
So now we have an exponent that's
under-- now we have a fraction that's being raised to the x.
Well again the same thing, we're going
to want to use our negative exponent role.
So I'm going to rewrite this as 2
to the negative first power times x equals 32.
I multiply these, 2 the negative x.
Now again I want to think about 32.
Now a lot of times people sometimes kind of get confused.
What we want to do is we know we can't rewrite 2 as a base 32.
So we want to rewrite 32 as a base 2.
So you always want to kind of go back to your smaller base.
And so you're trying to see, can I take that smaller base
and raise it to a power to get to my other base?
So a lot of times what I just tell students to do
is just to the side, just start writing down what each of them
equal.
2 to the first power is 2.
2 squared is 4.
2 cubed is 8.
2 to the fourth power is 16.
2 to the fifth power is 32.
So therefore I know 32 is equivalent to 2
to the fifth power.
Now I have my one to one property.
So I can say negative x is equal to 5.
Just divide by a negative 1 on both sides.
x equals a negative 5.
In the next example, you can see now we
have a fraction on both sides.
And again, but we want to kind of look at-- fractions a lot
of times, students just freeze up.
They see a fraction, they freeze up.
But again, we can break down the fraction
into the numerator and the denominator
and see how the numerator and the denominator
are related on both sides.
For my numerators, you can see just 2 to 4 is being squared.
From 3 to 9 is just being squared, right?
So all I can do is I can rewrite this as 2/3
to the x equals 2/3 squared.
Because remember when you're squaring a fraction,
you square the numerator and you square the denominator.
Or whenever you have a fraction raised to a power,
that power distributes to the numerator as well as
the denominator.
So therefore now they have the same base,
and I can say x is equal to 2.
And that rule always looks like this, a over b
to the m equals a to the m over b to the m.
So those are kind of like the simple ones,
just kind of looking at.
Usually what you're going to do is find the lower base
and then rewrite to the other base.
But now what we're going to do is
going to get into some problems that
are going to be a little bit more difficult where
the base is not so much apparent.
Because in this case, I saw my lower number was 4
and I said that's my base.
Can I rewrite 16 as a base 4?
And we could.
Well here my lower base is 9.
Can I rewrite 27 raised to the ninth power?
And I think crap, no I can't.
9 squared is 81.
So I'm kind of stuck there.
So now what I need to do is think about a lower base
that I can raise to give me 9, as well as a lower base that
can give me 27.
Well automatically when I see 9 I think of 3 squared.
So then I say all right, well if I know
3 squared, 3 to the first is 3.
3 squared is 9.
Hopefully you know what 3 cubed is,
but if you don't, you'll figure out that it's going to be 27.
So therefore I actually can rewrite both
of these with the same base 3.
Whereas this one is going to be 3 squared x.
And this one is 3 cubed.
Actually I don't really need my parentheses.
Now I apply my power rule.
So I get 3 to the 2x equals 3 cubed.
Therefore I have 2x is equal to 3.
Divide by 2, divide by 2.
x equals 3/2.
In this next case, now we just have
an exponent that's not just-- before I've been using just x.
And it's been pretty simple, right?
But now we can also have expressions that are exponents.
But it's not really going to matter as much.
Again, in this case I see 2 and I see 8.
I know that I can raise 2 to the third power
and that's going to give me 8.
So I just have 2 raised to the 2x minus 1 equals 2 cubed.
Now again, whenever your bases are exactly the same,
the powers are just equal to each other.
So now I have a two step equation
that I can just solve based on my understanding of solving
equations.
I'm getting a lot of answers that are 2, aren't I?
And there you go.
Now in this case again we look at 125 and 5.
And automatically when I see 25, I think oh OK, well again,
looking for the change of base, not every problem
that you do solving equations is going
to be using the change of base.
But any time you see a square number,
think to use the one to one property first.
I see 25.
I think 5 squared.
That's a square number.
Now is 125, can I also take 5 and raise it to a power
and give me 125?
Yes, it's going to be 3.
So therefore I'll rewrite this as 5 cubed times 2x plus 1
equals 5 squared.
Now to multiply these, I have to make sure
that I apply the distributive property.
Because remember using the power rule,
you're applying the distributive property.
Bases are the same.
So when I apply the distributive property,
I'm going to distribute.
So therefore I'll have 6x plus 3 equals 2 minus 3 minus 3.
6x equals negative 1.
Divide by 6, divide by 6.
x equals a negative 1/6.
That doesn't look good.
And then last but not least, we could also
have our powers be expressions on both sides.
Again I look at 8 and 16 and I think of 16 as being 4 squared,
right?
But the problem is I can't write 4 raised to a power
to give me 8.
Because if you think about it, you
have 4 to the first power, which is 4.
4 squared is equal to 16.
So we're kind of stuck.
We know 16 is 4 squared.
But we can't write 8 as 4 to a power.
So we have to look at a smaller number.
And then fortunately I used a lot of these to be simple,
would be 2.
Notice how 2 raised to a power gives us 16 as well as 8.
So that's what I'm going to want to use.
And again ladies and gentlemen, the main important thing
is I would recommend knowing 2, 3 at least
to the first couple of powers, because when
you're using the one to one property, most
of the problems in the book or you're seeing on tests,
you're using very similar numbers.
Even when it's a fraction, the numbers in the top and bottom
are going to be square numbers raised to a power.
So therefore 2 to the eighth is 2 cubed,
times x plus 3 equals 2 to the fourth power,
times-- oops-- 2 to the fourth power times x minus 1.
Now I need to apply the distributive property.
So it's 2 to 3 times x plus 1 equals 2
to the 4 times x minus 1.
Now I can get rid of my bases.
And I'm left with 3x plus 3 equals 4x minus 4.
Now I just need to simply solve for x.
So I'm going to get the x's on the same side.
I'm going to get 7 equals x or x equals 7.
So there you go ladies and gentlemen.
That is how you solve an exponential question using
the one to one property.
Thanks.