CHAPTER 5: Exponential and Logarithmic Functions

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1 MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions and Graphs 5.3 Logarithmic Functions and Graphs 5.4 Properties of Logarithmic Functions 5.5 Solving Exponential and Logarithmic Equations 5.6 Applications and Models: Growth and Decay; and Compound Interest 5.2 Exponential Functions and Graphs Graph exponential equations and exponential functions. Solve applied problems involving exponential functions and their graphs. Exponential Function The function f(x) = a x, where x is a real number, a > 0 and a 1, is called the exponential function, base a. The base needs to be positive in order to avoid the complex numbers that would occur by taking even roots of negative numbers. The following are examples of exponential functions: Graphing Exponential Functions To graph an exponential function, follow the steps listed: 1. Compute some function values and list the results in a table. 2. Plot the points and connect them with a smooth curve. Be sure to plot enough points to determine how steeply the curve rises. 1

2 Graph the exponential function y = f (x) = 2 x. As x increases, y increases without bound. As x decreases, y decreases getting close to 0; as x, y 0. The x axis, or the line y = 0, is a horizontal asymptote. As the x inputs decrease, the curve gets closer and closer to this line, but does not cross it. Graph the exponential function This tells us the graph is the reflection of the graph of y = 2 x across the y axis. Selected points are listed in the table. As x increases, the function values decrease, getting closer and closer to 0. The x axis, y = 0, is the horizontal asymptote. As x decreases, the function values increase without bound. 2

3 Graphs of Exponential Functions Observe the following graphs of exponential functions and look for patterns in them. Graph y = 2 x 2. The graph is the graph of y = 2 x shifted to right 2 units. For the base between 0 and 1, the graph goes DOWN toward the x axis to the right. For the base between greater than 1, the graph goes UP to the right. Graph y = x. The graph y = 2 x is a reflection of the graph of y = 2 x across the y axis; y = 2 x is a reflection across the x axis; y = 2 x + 5 or y = 5 2 x is a shift up 5 units. Graph 3 Application The amount of money A that a principal P will grow to after t years at interest rate r (in decimal form), compounded n times per year, is given by the formula Graph 1 Graph 2 y = 5 H.A. Graph 4 1 3

4 Suppose that $100,000 is invested at 6.5% interest, compounded semiannually. a) Find a function for the amount to which the investment grows after t years. b) Graph the function. c) Find the amount of money in the account at t = 0, 4, 8, and 10 yr. d) When will the amount of money in the account reach $400,000? Solution: a) Since P = $100,000, r = 6.5%=0.65, and n = 2, we can substitute these values and write the following function b) Use the graphing calculator with viewing window [0, 30, 0, 500,000]. Solution continued: c) We can compute function values using function notation on the home screen of a graphing calculator. Solution continued: c) We can also calculate the values directly on a graphing calculator by substituting in the expression for A(t): 4

5 Solution continued: d) Set 100,000(1.0325) 2t = 400,000 and solve for t, which we can do on the graphing calculator. Graph the equations y 1 = 100,000(1.0325) 2t y 2 = 400,000 Then use the intersect method to estimate the first coordinate of the point of intersection. Solution continued: d) Or graph y 1 = 100,000(1.0325) 2t 400,000 and use the Zero method to estimate the zero of the function coordinate of the point of intersection. Regardless of the method, it takes about years, or about 21 yr, 8 mo, and 2 days to reach $400,000. The Number e e is a very special number in mathematics. Leonard Euler named this number e. The decimal representation of the number e does not terminate or repeat; it is an irrational number that is a constant; e» Find each value of e x, to four decimal places, using the e x key on a calculator. a) e 3 b) e 0.23 c) e 0 d) e 1 Solution: a) e b) e c) e 0 = 1 d) e

6 Graphs of Exponential Functions, Base e Graph f (x) = e x and g(x) = e x. Use the calculator and enter Y1 = e x and Y2 = e x. Enter numbers for x. Graphs of Exponential Functions, Base e The graph of g is a reflection of the graph of f across the y axis. Graph f (x) = e x + 3. Solution: The graph f (x) = e x + 3 is a translation of the graph of y = e x left 3 units. Graph f (x) = e 0.5x. Solution: The graph f (x) = e 0.5x is a horizontal stretching of the graph of y = e x followed by a reflection across the y axis. When x = 1 on the left and x = 2 on the right, we get the same y value because 0.5( 2) = 1. 6

7 403/4. Find each of the following, to four decimal places, using a calculator. Graph f (x) = 1 e 2x. Solution: The graph f (x) = 1 e 2x is a horizontal shrinking of the graph of y = e x followed by a reflection across the y axis and then across the x axis, followed by a translation up 1 unit. 403/14. Graph the function by substituting and plotting points. Then check your work using a graphing calculator: f(x) = 3 x 403/10. Match the function with one of the graphs: f(x) = 1 e x 7

8 403/20. Graph the function by substituting and plotting points. Then check your work using a graphing calculator: f(x) = 0.6 x 3 403/34. Sketch the graph of the function and check the graph with a graphing calculator. Describe how the graph can be obtained from (4 x) the graph of a basic exponential function: f(x) = 3 8

9 404/50. Use the compound interest formula to find the account balance A with the given conditions: A = account balance; t = time, in years; n = number of compounding periods per year; r = interest rate; P = principal P = $120,000 at 2.5% for 10 years; quarterly 405/58. Growth of Bacteria Escherichia coli. The bacteria Escherichia coli are commonly found in the human intestines. Suppose that 3000 of the bacteria are present at time t = 0. Then under certain conditions, t minutes later, the number of bacteria present is N(t) = 3000(2) t/20. a) How many bacteria will be present after 10 min? 20 min? 30 min? 40 min? 60 min? b) Graph the function. c) These bacteria can cause intestinal infections in humans when the number of bacteria reaches 100,000,000. Find the length of time it takes for an intestinal infection to be possible. 407/82. Use a graphing calculator to match the equation with one of the figures (a) (n): f(x) = (e x + e x ) / 2 407/76. Use a graphing calculator to match the equation with one of the figures (a) (n): y = 2 x + 2 x 9

10 407/84. Use a graphing calculator to find the point(s) of 407/88. Solve graphically: e x = x 3 intersection of the graphs of each of the following pairs of equations: y = 4 x + 4 x and y = 8 2x x 2 10