Exponential Functions

1. Exponential Functions By Dr. Julia Arnold and Ms. Karen Overman using Tan’s 5th edition Applied Calculus for the managerial , life, and social sciences text

2. The ExponentialFunction The exponential function with base a is denoted by f(x) = ax , where a > 0, a 1 and x represents any real number. Note: a is a positive constant and the variable x appears in the exponent, which is why the function is given the name exponential function.

3. Let’s look at some examples of exponential functions. Example 1: Graph using a table of values. x 2x 0 1 1 2 2 4 -1 1/2 -2 1/4

4. Let’s look at another example. Example 2: Graph using a table of values. X 1 2 0 -1 -2 1/3 1/9 1 3 9

5. Now let’s look at some of the properties of exponential functions. • 1. The domain is all real numbers. • 2. The range is (0, ). In other words for any x value the • function value is positive!! • 3. The graph passes through the point (0, 1) since any • positive number raised to the zero power is 1. • 4. The function is increasing if a > 1 and the function is • decreasing if 0 < a < 1.

6. Another exponential function, f(x) =ex is called the natural exponential function. e is called a transcendental number which means it is not the root of another number. On the graphing calculator, you can find e by pushing the yellow 2nd function button and the LN key. On the display you will see e^( ,type 1 and close the parenthesis. You should see e^( 1). Press the enter key and you will see 2.718281828 which represents an approximation of e.

7. To graph f(x) =ex , you will need to use your calculator. Graph f(x) =exusing a table of values. • x ex • e1 = 2.72 • e2 = 7.39 • 0 e0 = 1 • -1 e-1 = 0.37 • -2 e-2 = 0.14

8. Let’s graph one more exponential for practice. Graph f(x) = 4x . First you should make a table of values. Then plot the points and connect the points with a smooth curve. x 4x -2 -1 0 1 2 1/16 ¼ 1 4 16

9. Solving equations involving exponentials. We are going to look at some equations involving exponential functions. Before we do that let’s review the rules for exponents. • Rules for exponents. • ax ay = ax+y 2. ax /ay = ax-y • 3. (ax )y = axy 4. (ab)x = ax bx • 5. (a/b)x = ax /bx 6. a0 = 1 • 7. a-n = 1/ an

10. Solving equations involving exponentials. Example 1: Solve for x. 2x =32 Since, 32 can be written as 25, 2x = 25 and for the two sides to be equal their exponents must be equal. Thus, x = 5.

11. Solving equations involving exponentials. Example 2: Solve for x. 9x = 1/27 Now, 27 is not 9 raised to a power, but both are powers of 3: 9 = 3² and 27 = 3³. Using rules 3 and 7 for exponents you can rewrite each side so that each is 3 raised to a power. 9x = 1/27 Rewrite 9 and 27 as 3 to a power: (3²) x = 1/(3³) Use the rules for exponents to simplify: 3 2x = 3-3 For the two sides to be equal their exponents must be equal. Thus, 2x = -3 or x = -3/2.

12. Let’s review. 1. An exponential function has a positive constant base (other than 1) and a variable exponent. 2. To graph an exponential you can use a table of values. Then plot the points and connect them with a smooth curve. 3. To solve an exponential equation, rewrite each side so that they have the same base. Then set the exponents equal and solve for the variable. Note: In the next section we will solve more exponential equations and we will look at the case where we cannot rewrite the two sides so that they have the same base.