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.

Actually your answer to the last problem is wrong. The answer should be 13 instead of 7. You made a mistake by writing 3(x + 1) instead of 3(x + 3).

soooooooo useful i will access your video's more often due to there pure awesomeness

the final example became wrong lol

I’m in grade 9 doing a Cambridge system and i always get confused with these equations and this video was extremely helpful. Thank you so much

Thanks for the this ways it's very easy

The last one should be 3x + 9 instead of 3x + 3 giving u 13 as the finial answer