Slide 1: Copyright © Cengage Learning. All rights reserved. 3 Exponential and Logarithmic Functions
Slide 2: 3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Copyright © Cengage Learning. All rights reserved.
What You Should Learn: Recognize and evaluate logarithmic functions with base a . Graph logarithmic functions. Recognize, evaluate, and graph natural logarithmic functions. Use logarithmic functions to model and solve real-life problems. What You Should Learn
Slide 4: Logarithmic Functions
Logarithmic Functions: Logarithmic Functions Every function of the form f ( x ) = a x passes the Horizontal Line Test and therefore must have an inverse function. This inverse function is called the logarithmic function with base a .
Logarithmic Functions: Logarithmic Functions The equations y = log a x and x = a y are equivalent. The first equation is in logarithmic form and the second is in exponential form. For example, the logarithmic equation 2 = log 3 9 can be rewritten in exponential form as 9 = 3 2 . The exponential equation 5 3 = 125 can be rewritten in logarithmic form as log 5 125 = 3.
Logarithmic Functions: Logarithmic Functions When evaluating logarithms, remember that a logarithm is an exponent . This means that log a x is the exponent to which a must be raised to obtain x . For instance, log 2 8 = 3 because 2 must be raised to the third power to get 8.
Example 1 – Evaluating Logarithms: Example 1 – Evaluating Logarithms Use the definition of logarithmic function to evaluate each logarithm at the indicated value of x . a. f ( x ) = log 2 x , x = 32 b. f ( x ) = log 3 x , x = 1 c. f ( x ) = log 4 x , x = 2 d. f ( x ) = log 10 x , x = Solution: a. f ( 32 ) = log 2 32 because 2 5 = 32. = 5 b. f ( 1 ) = log 3 1 because 3 0 = 1. = 0
Example 1 – Solution: Example 1 – Solution c. f ( 2 ) = log 4 2 because 4 1/2 = = 2. = d. because . cont’d
Logarithmic Functions: Logarithmic Functions The logarithmic function with base 10 is called the common logarithmic function. It is denoted by log 10 or simply by log. On most calculators, this function is denoted by . The following properties follow directly from the definition of the logarithmic function with base a .
Slide 11: Graphs of Logarithmic Functions
Graphs of Logarithmic Functions: Graphs of Logarithmic Functions To sketch the graph of y = log a x , you can use the fact that the graphs of inverse functions are reflections of each other in the line y = x .
Example 5 – Graphs of Exponential and Logarithmic Functions: Example 5 – Graphs of Exponential and Logarithmic Functions In the same coordinate plane, sketch the graph of each function. a. f ( x ) = 2 x b. g ( x ) = log 2 x
Example 5(a) – Solution: Example 5(a) – Solution For f ( x ) = 2 x , construct a table of values. By plotting these points and connecting them with a smooth curve, you obtain the graph shown in Figure 3.14. Figure 3.14
Example 5(b) – Solution: Example 5(b) – Solution Because g ( x ) = log 2 x is the inverse function of f ( x ) = 2 x , the graph of g is obtained by plotting the points ( f ( x ), x ) and connecting them with a smooth curve. The graph of g is a reflection of the graph of f in the line y = x , as shown in Figure 3.14. cont’d Figure 3.14
Graphs of Logarithmic Functions: Graphs of Logarithmic Functions The nature of the graph in Figure 3.15 is typical of functions of the form f ( x ) = log a x , a 1. They have one x -intercept and one vertical asymptote. Notice how slowly the graph rises for x 1. Figure 3.15
Graphs of Logarithmic Functions: Graphs of Logarithmic Functions The basic characteristics of logarithmic graphs are summarized in Figure 3.16. Graph of y = log a x , a 1 • Domain: (0, ) • Range: ( , ) • x -intercept: (1, 0) • Increasing • One-to-one, therefore has an inverse function Figure 3.16
Graphs of Logarithmic Functions: Graphs of Logarithmic Functions • y -axis is a vertical asymptote (log a x → as x → 0 + ). • Continuous • Reflection of graph of y = a x about the line y = x. The basic characteristics of the graph of f ( x ) = a x are shown below to illustrate the inverse relation between f ( x ) = a x and g ( x ) = log a x . • Domain: ( , ) • Range: (0, ) • y -intercept : (0, 1) • x -axis is a horizontal asymptote ( a x → 0 as x → ).
Slide 19: The Natural Logarithmic Function
The Natural Logarithmic Function: The Natural Logarithmic Function We will see that f ( x ) = e x is one-to-one and so has an inverse function. This inverse function is called the natural logarithmic function and is denoted by the special symbol ln x , read as “the natural log of x ” or “el en of x. ” Note that the natural logarithm is written without a base. The base is understood to be e .
The Natural Logarithmic Function: The Natural Logarithmic Function The definition above implies that the natural logarithmic function and the natural exponential function are inverse functions of each other. So, every logarithmic equation can be written in an equivalent exponential form, and every exponential equation can be written in logarithmic form. That is, y = In x and x = e y are equivalent equations.
The Natural Logarithmic Function: The Natural Logarithmic Function Because the functions given by f ( x ) = e x and g ( x ) = In x are inverse functions of each other, their graphs are reflections of each other in the line y = x . This reflective property is illustrated in Figure 3.19. On most calculators, the natural logarithm is denoted by , as illustrated in Example 8. Reflection of graph of f ( x ) = e x about the line y = x . Figure 3.19
Example 8 – Evaluating the Natural Logarithmic Function: Example 8 – Evaluating the Natural Logarithmic Function Use a calculator to evaluate the function given by f ( x ) = In x for each value of x . a. x = 2 b. x = 0.3 c. x = –1 d. x = 1 +
Example 8 – Solution: Example 8 – Solution Function Value Graphing Calculator Display Keystrokes
The Natural Logarithmic Function: The Natural Logarithmic Function The four properties of logarithms are also valid for natural logarithms.
Slide 26: Application
Example 11 – Human Memory Model: Example 11 – Human Memory Model Students participating in a psychology experiment attended several lectures on a subject and were given an exam. Every month for a year after the exam, the students were retested to see how much of the material they remembered. The average scores for the group are given by the human memory model f ( t ) = 75 – 6 In( t + 1), 0 t 12, where t is the time in months.
Example 11 – Human Memory Model: Example 11 – Human Memory Model a. What was the average score on the original ( t = 0) exam? b. What was the average score at the end of t = 2 months? c. What was the average score at the end of t = 6 months? Solution: a. The original average score was f ( 0 ) = 75 – 6 ln( 0 + 1) = 75 – 6 ln 1 cont’d Substitute 0 for t . Simplify.
Example 11 – Solution: Example 11 – Solution = 75 – 6(0) = 75. b. After 2 months, the average score was f ( 2 ) = 75 – 6 ln( 2 + 1) = 75 – 6 ln 3 75 – 6(1.0986) 68.4. cont’d Property of natural logarithms Solution Substitute 2 for t . Simplify. Use a calculator. Solution
Example 11 – Solution: Example 11 – Solution c. After 6 months, the average score was f ( 6 ) = 75 – 6 ln( 6 + 1) = 75 – 6 ln 7 75 – 6(1.9459) 63.3. cont’d Substitute 6 for t . Simplify. Use a calculator. Solution