Today we're going to talk about exponential equations. So we'll start by giving a general form for the equation of an exponential function. And then we'll talk about the characteristics of exponential functions, including the domain and the range.
So let's talk about exponential equations. Exponential equations, as the name implies, involve exponents, but in a different way than we've seen in something like a polynomial equation which might have an expression x squared in it. So in an exponential equation, the variable is in our exponent instead of in the base. So our general form for an exponential equation is y is equal to a times b to the x, where b is the base of our variable exponent and a is the constant multiplier.
The domain of an exponential function is all real numbers, so we have some restrictions on our base b to maintain our property of an all-real-number domain. So the first restriction is that the base b cannot be negative. Having a negative number for our base is going to exclude values from our domain. For example, if I have a base of negative 16 to the exponent 1/2, using my property of fractional exponents and radicals, I know I can rewrite this as the square root of negative 16, which is just going to give me 4i, which is an imaginary number. And that's because a negative number underneath an even root radical is always going to lead to an unreal solution.
The second restriction on our base is that it cannot be equal to 0. So a base being equal to 0 would exclude all of the negative values of x from our domain. So we restrict our base to not being equal to 0 and also not being negative.
And the third restriction then is that the base should not be equal to 1. And this does not exclude values from our domain. But it would be to a linear function. If our base was 1, 1 to any exponent is still just going to be equal to 1. And when we multiply that by our value for a, that's just going to give us a constant number. So we'd have an equation y is equal to some constant value, which is just a linear equation.
So let's look more closely at an example of an exponential equation. And we can draw some conclusions about its domain and range. So I've got the exponential equation, y equals 2 to the x. And I know it's exponential because I have a variable in my exponent.
I'm going to evaluate it for different values of x by substituting an x value into the equation for x to determine y. So if x is 1, this would become 2 to the first power, which is just equal to 2. So when x is 1, y equals 2.
I'll do the same thing when x is 2. This would become 2 to the second power, which is just 4. When x is 3, then y equals 2 to the third power, which is 8. And when x is 4, then y will equal 2 to the fourth power, which is 16.
So I can see that as my x values are increasing, my y values are also increasing but at an increasing rate. It's getting bigger faster and faster.
I'll do the same thing using x values that are negative. So when I have a negative 1 for my x value, y will be equal to 2 to the negative 1, which is to say as 1 over 2 to the positive 1, which is 1 over 2, or 0.5. When x is equal to negative 2, this would be 2 to the negative 2, which would give me 1 over 2 to the positive 2, or 1 over 4, which is 0.25.
When x is negative 3, I have 2 to the negative third, which is 1 over 2 to the positive third, or 1 over 8, which is 0.125. And when x is negative 4, that's going to give me 1 over positive 16, which is 0.0625. So we can see here that as x is decreasing, then our values for y are getting closer and closer to 0.
So thinking about our domain, we can say that our domain is going to be all real x values. And our range is going to be all real y values greater than 0. So we have y values that are positive going towards positive infinity. And we have y values that are approaching 0 but never actually equaling 0.
So finally, let's look at one more example of an exponential function, this time written in function notation. And we'll evaluate it for different values of x. So I'm first going to use a value of x of 0. So I'm going to input that for x into my equation.
And this will become 2 times 3 to the 0 power. So we have to think about the order of operations. We know that we have to evaluate the exponent first and then multiply by the number in front.
So first we'll do 3 to the 0 power, which is just 1. Anything to the 0 power is 1. So we have 2 times 1, which is just going to be 2. So when x is 0, f of x or my output is 2.
Let's try the same thing with the input of 2. So this becomes f of 2 equals 2 times 3 to the second power. Again, we'll start by evaluating our exponent. 3 to the second power gives me 9. So this is 2 times 9, which is 18. So my value when my input is 2, my output is 18.
Finally, let's try input of 4. So this is 2 times 3 to the fourth power. We'll first do 3 to the fourth, which is 81. So this becomes 2 times 81, which would be 162.
So we can see that when we have a exponential function that is increasing, it's increasing very quickly. This is a big characteristic of exponential functions. We're increasing from 2 to 18 and then from 18 to 162. So it gets very big very quickly.
So let's go over our key points from today. An exponential equation is an equation involving a constant value raised to a variable power. The general form is a times b to the x.
The domain of an exponential function is all x values. The range is all y values greater than 0 for a greater than 0, and all y values less than 0 for a less than 0. Exponential functions as compared to linear functions have graphs that increase at an increasing rate or decrease at a decreasing rate.
So I hope that these key points and examples helped you understand a little bit more about exponential equations. Keep using your notes. And keep on practicing. And soon you'll be a pro. Thanks for watching.