Hi, and welcome. My name is Anthony Varela. And today I'm going to introduce exponential equations. So we'll compare this to a polynomial equation, talk about some restrictions to the base, and then evaluate exponential functions.
So first, I'd like to compare exponential equations to polynomial equations. So here's a polynomial equation, y equals 2x squared plus 5x minus 3. We see variables and we see exponents, but this is not an exponential equation. An exponential equation is an equation involving a constant value raised to a variable power.
So for example, y equals a times b to the power of x is our general exponential equation. We have a base number b that is being raised to a variable power x. And all of this is being multiplied then by a.
Note, the domain of exponential equations is all real numbers. So x could take on any real value. And because of that, there are some restrictions to the base, what the base can and cannot be.
So the first restriction that I could talk about is that the base must not be negative. And this is because negative bases exclude certain values of x. So there are certain cases where a negative value raised to a power is OK. And there are certain values where it's not OK.
So for example, negative 5 raised to the power of negative 1/3 evaluates to a real number. That's OK. But for example, if we have negative 5 raised to the power of negative 1/2, I can think of this as being 1 divided by the square root of negative 5.
And the square root of a negative number isn't real. So we cannot have bases that are negative because certain values of x would result in not real numbers.
Also, the base cannot equal 0. And this is because this excludes non-positive values of x. So for example, if we have a base of 0 that we raise to a positive power, that's OK. That equals 0.
But if we have y equals 0 to the power of negative 2, I can think of this as 1 over 0 squared. And this would be division by 0. So our base cannot be negative and it cannot be 0. So we say that the base must be positive or greater than 0.
Now there is another restriction, that the base cannot equal 1. And this isn't because it restricts any values of x. It's just because with a base of 1 we would have a linear relationship.
So let's assume that our a value is 1. So if we have a base of 1 as well, that means you would have 1 to the third power, for example. That equals 1.
If our exponent is negative, y would still equal 1. And if our exponent was 0, y would still equal 1, whether our exponent is positive, negative, or 0. This would be a linear relationship. There is no exponential behavior about it at all. So our base restrictions-- base must be a positive number. And it should not or cannot equal 1.
Well, now let's talk more about exponential behavior. So we're going to look at the equation y equals 2 raised to the power of x. So we're going to look at some positive values of x.
So when x equals 1, we'll evaluate 2 to the first power. When x equals 2, we'll evaluate 2 squared. For x equals 3, we'll evaluate 2 cubed. And when x equals 4, we'll evaluate 2 to the fourth power.
So 2 to the first power is 2. 2 squared is 4. 2 cubed is 8. And 2 to the fourth power is 16. So we can see then that as x increases by 1, we're doubling our y value.
And that's because our base here is 2. So for this equation, we can say that as x approaches positive infinity, y approaches positive infinity as well.
Well, now let's take a look at some negative values of x. So we're going to evaluate 2 to the negative first power, 2 to the power of negative 2, 2 to the power of negative 3, and 2 to the power of negative 4.
Evaluating 2 to the power of negative 1, this is 1 over 2, so 1/2. Evaluating 2 to the power of negative 2, this would be 1 over 2 squared, or 1/4. And 2 to the power of negative 3 is 1 over 2 cubed, or 1/8. And finally, 2 to the power of negative 4 is 1 over 2 to the fourth power, or 1/16.
So as x decreases by 1, we're cutting our y value in half. So for this equation, as x approaches negative infinity, what happens to our y value? Well, it goes down. It approaches 0.
It's never going to equal exactly 0 because in theory there's always something to cut in half, even though it's a very, very small amount. And it's never going to reach a negative value, so y approaches to 0. And this is characteristic when our a value is a positive number and if our base is greater than 1.
So I'd like to wrap up by evaluating some exponential functions. So we're going to be using this function, f of x equals 2 times 3 to the power of x. And we're going to evaluate f of 1, f of 2, and f of 3.
And the big idea here is that we need to apply that exponent first. So we're following the order of operations, applying the exponent, and then we'll multiply by 2.
So thinking about f of 1, I'm going to multiply 2 by 3 to the first power. So that would be 3. So 2 times 3 is 6.
To evaluate f of 2, I'm going to multiply 2 by 3 squared. So 2 times 9. And that gives us 18.
And to evaluate f of 3, I'm going to multiply 2 by 3 cubed. So three cubed is 27. I'm going to multiply that by 2 to get 54. So f of 3 is 54.
So let's review our introduction to exponential equations. Our general exponential equation is y equals a times b to the x. So we have a base number raised to a variable power x. And that's multiplied by an a value.
And we have some restrictions to our base. The base number must be a positive number. And it cannot equal 0.
We looked at some exponential behaviors, specifically when our a value was positive and our base value was greater than 1. And when evaluating exponential functions, we apply that exponent first, and then multiply by our a value.
So thanks for watching this tutorial on an introduction to exponential equations. Hope to see you next time.
