Subscribe! http://www.freemathvideos.com Want more math video lessons? Visit my website to view all of my math videos organized by course, chapter and section. The purpose of posting my free video tutorials is to not only help students but allow teachers the resources to flip their classrooms and allow more time for teaching within the classroom. Please feel free to share my resources with those in need or let me know if you need any additional help with your math studies. I am a true educator and here to help you out. 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.
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