Logarithmic Functions

Transcription

1 C H A P T ER Logarithmic Functions The human ear is capable of hearing sounds across a wide dynamic range. The softest noise the average human can hear is 0 decibels (db), which is equivalent to a mosquito flying three meters away. By comparison, a pop music concert, at about 115 db, is over 10 billion times louder. The decibel scale is an example of a logarithmic scale, which can be used to plot very large and very small numbers along a single axis. You will learn about logarithms and their uses..1 What s the Inverse of an Exponent? Logarithmic Functions as Inverses p Do I Have the Right Form? Exponential and Logarithmic Forms p It s All in the Graph Graphs of Logarithmic Functions p. 93 Transformers Again! Transformations of Logarithmic Functions p. 301 Chapter l Logarithmic Functions 9

2 80 Chapter l Logarithmic Functions

3 .1 What s the Inverse of an Exponent? Logarithmic Functions as Inverses Objectives In this lesson you will: l Graph the inverse of exponential functions. l Define the inverse of exponential functions. l Determine the domain, range, and asymptotes of the inverse of exponential functions. Key Terms l logarithm l logarithmic function l common logarithm l natural logarithm Problem 1 Graphing the Inverse of Exponential Functions 1. Graph and label the function f(x) x and the line y x. y x Lesson.1 l Logarithmic Functions as Inverses 81

4 . Complete the tables for the function f(x) x and its inverse. Then plot each point on the grid from Question 1. x f( x) x x f 1 ( x) Connect the points of f 1 (x) with a smooth curve. Then, label the graph as f 1 (x). 4. Is the inverse f 1 (x) a function? Explain. 5. What are the domain and range of the exponential function? 6. What are the domain and range of the inverse of the exponential function? 8 Chapter l Logarithmic Functions

5 . What do you notice about the domain and range of the exponential function and its inverse? 8. What is the asymptote of the exponential function? 9. What is the asymptote of the inverse of the exponential function? 10. What do you notice about the asymptotes of the exponential function and its inverse? 11. Graph and label the function g(x) 3 x and the line y x. 8 6 y x 6 8 Lesson.1 l Logarithmic Functions as Inverses 83

6 1. Complete the tables for the function g(x) 3 x and its inverse. Then plot each point on the grid from Question 11. x g( x) 3 x 3 1 x g 1 ( x) Connect the points of g 1 (x) with a smooth curve. Then, label the graph as g 1 (x). 14. Is the inverse g 1 (x) a function? Explain. 15. What are the domain and range of the exponential function? 16. What are the domain and range of the inverse of the exponential function? 84 Chapter l Logarithmic Functions

7 1. What do you notice about the domain and range of the exponential function and its inverse? 18. What is the asymptote of the exponential function? 19. What is the asymptote of the inverse of the exponential function? 0. What do you notice about the asymptotes of the exponential function and its inverse? 1. The graph of the function h( x) b x and the line y x are shown. Sketch the graph of h 1 ( x). y h(x) y = x x Lesson.1 l Logarithmic Functions as Inverses 85

8 . Is the inverse h 1 (x) a function? Explain. 3. What are the domain and range of the exponential function? 4. What are the domain and range of the inverse of the exponential function? 5. What do you notice about the domain and range of the exponential function and its inverse? 6. What is the asymptote of the exponential function?. What is the asymptote of the inverse of the exponential function? 8. What do you notice about the asymptotes of the exponential function and its inverse? 86 Chapter l Logarithmic Functions

9 Problem Defining the Inverse of an Exponential Function In Problem 1, you graphed the inverse of the exponential function h(x) b x as a reflection about the line y x. You also examined the domain, range, and asymptote of the exponential function and its inverse. You have not encountered a function with the properties of the inverse of an exponential function. So, it is necessary to define a new function for the inverse of an exponential function. The logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce the number. For example, if a b c, then the logarithm of a to the base b is c. This logarithm is written as log b a c. A logarithmic function is a function involving a logarithm. Logarithms were first conceived by a Swiss clockmaker and amateur mathematician Joost Bürgi but became more widely known and used after the publication of a book by Scottish mathematician John Napier in Logarithms were originally used to make complex computations in astronomy, surveying, and other sciences easier and more accurate. With the invention of calculators and computers, the use of logarithms as a tool for calculation has decreased. However, many real-world situations can be modeled using logarithmic functions. The two frequently used logarithms are logarithms with base 10 and base e. A common logarithm is a logarithm with base 10 and is usually written log without a base specified. A natural logarithm is a logarithm with base e, Euler s constant, and is usually written as In. Many graphing calculators only have keys for common logarithms and natural logarithms. 1. Graph and label the functions f (x) log x and g(x) In x using a graphing calculator. Lesson.1 l Logarithmic Functions as Inverses 8

10 . What is the inverse of the logarithmic function f(x) log x? 3. What is the inverse of the logarithmic function f(x) In x? Be prepared to share your methods and solutions. 88 Chapter l Logarithmic Functions

11 . Do I Have the Right Form? Exponential and Logarithmic Forms Objectives In this lesson you will: l Convert from exponential equations to logarithmic equations. l Convert from logarithmic equations to exponential equations. l Evaluate logarithmic expressions. Problem 1 Converting Between Exponential and Logarithmic Forms Remember that the definition of a logarithm allows you to convert the exponential equation a b c to the logarithmic equation log b a c. 1. Write each exponential equation as a logarithmic equation using the definition of logarithms. a. 3 8 b c ,000 d e. ( 1 3 ) 4 81 f. ( 1 5 ) Write each logarithmic equation as an exponential equation using the definition of logarithms. a. log 16 4 b. log c. log d. log a b c Lesson. l Exponential and Logarithmic Forms 89

12 Problem Evaluating Logarithmic Expressions 1. Evaluate each logarithmic expression without using a calculator. Explain how you calculated each. a. log 104 b. log c. log 343 d. log 100,000 e. log 4 f. log 3 1 g. log h. log 100, Chapter l Logarithmic Functions

13 . Evaluate each logarithmic expression. Use a calculator if necessary. a. log 100 b. log 0.01 c. log d. log 0 e. log 00 f. log 0.0 g. log Evaluate each logarithmic expression. Use a calculator if necessary. a. In 100 b. In 10 c. In 5 d. In 0.5 e. In f. In e g. In e 3 h. In 6.5 Lesson. l Exponential and Logarithmic Forms 91

14 4. Evaluate each logarithmic expression. Use a calculator if necessary. a. log 64 b. log 8 c. log 8 d. In 1 e. In 4 f. In 3 g. log 36 h. log 6 i. In 5 3 j. In 5 k. In 3 Be prepared to share your methods and solutions. 9 Chapter l Logarithmic Functions

15 .3 It s All in the Graph Graphs of Logarithmic Functions Objectives In this lesson you will: l Graph logarithmic functions. l Determine the characteristics of logarithmic functions. Problem 1 The Extraordinary Graph of an Exponential Function 1. Graph the function f (x) 10 x for x-values between 5 and 5 and y-values between 0 and 100. Lesson.3 l Graphs of Logarithmic Functions 93

16 . Graph the function f (x) 10 x for x-values between 10 and 10 and y-values between 0 and Graph the function f (x) 10 x for x-values between 0 and 0 and y-values between 0 and 10, Describe the similarities and differences between the graphs in Questions 1 through Describe how the scale of the x-axis changes from each graph. 94 Chapter l Logarithmic Functions

17 6. Describe how the scale of the y-axis changes from each graph.. Imagine a very large sheet of graph paper with every square grid measuring one tenth of an inch. You set the scale on both the x-axis and the y-axis at one unit. Describe the coordinates of each point on the graph of the function f (x) 10 x and the point s distance from the origin on the graph paper. a. A point one inch above the x-axis. b. A point one foot above the x-axis. c. A point one hundred feet above the x-axis. d. A point one mile above the x-axis. e. A point one foot to the right of the origin. As you can see in Problem 1, the y-values of an exponential function increase very rapidly, even for relatively small changes in the values of x. Lesson.3 l Graphs of Logarithmic Functions 95

18 Problem The Extraordinary Graph of a Logarithmic Function 1. Graph the function f (x) log x for x-values between 0 and 100 and y-values between 5 and 5.. Graph the function f (x) log x for x-values between 0 and 1000 and y-values between 10 and Chapter l Logarithmic Functions

19 3. Graph the function f (x) log x for x-values between 0 and 10,000 and y-values between 0 and Describe the similarities and differences between the graphs in Questions 1 through Describe how the scale of the x-axis changes from each graph. 6. Describe how the scale of the y-axis changes from each graph.. Imagine a very large sheet of graph paper with every square grid measuring one tenth of an inch. You set the scale on both the x-axis and the y-axis at one unit. Describe the coordinates of each point on the graph of the function f (x) log x and the point s distance from the origin on the graph paper. a. A point one inch along the x-axis. b. A point one foot along the x-axis. Lesson.3 l Graphs of Logarithmic Functions 9

20 c. A point one hundred feet along the x-axis. d. A point one mile along the x-axis. e. A point one foot above the x-axis. As you can see in Problem, the y-values of a logarithmic function increase very slowly, even for relatively large values of x. Problem 3 Logarithmic Functions 1. The graph of f (x) log a x is shown. Label the coordinates of each of the three points. 4 y f(x) = log a x x. What are the domain and range of f (x) log a x? 98 Chapter l Logarithmic Functions

21 3. What are the intercepts of f (x) log a x? 4. What are the asymptotes of f (x) log a x? 5. For what x-values is the function f (x) log a x increasing or decreasing? Be prepared to share your methods and solutions. Lesson.3 l Graphs of Logarithmic Functions 99

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23 .4 Transformers Again! Transformations of Logarithmic Functions Objective In this lesson you will: l Transform logarithmic functions algebraically and graphically. Problem 1 Horizontal and Vertical Translations Earlier you learned that the graph of a function f (x) is translated vertically k units if a constant k is added to the equation: f (x) k. 1. Sketch and label the graphs of f (x) log x, f (x) log x 3, and f (x) log x 4. Lesson.4 l Transformations of Logarithmic Functions 301

24 The graph of a function f (x) is translated horizontally h units if a constant h is subtracted from the variable in the function: f (x h).. Sketch and label the graphs of f (x) log x, f (x) log(x 3), and f (x) log (x 4). 3. Sketch and label the graph of f (x) In(x 3) 5 using the graph of f (x) In x. y f(x) = In x x Chapter l Logarithmic Functions

25 Problem Reflections Earlier you learned that the graph of a function f (x) is reflected about the x-axis if the equation of the function is multiplied by 1: f (x). 1. Sketch and label the graph of f (x) In x using the graph of f (x) In x. y f(x) = In x x 6 8 Earlier you learned that the graph of a function f (x) is reflected about the y-axis if the argument of the equation of the function is multiplied by 1: f ( x).. Sketch and label the graph of f (x) log( x) using the graph of f (x) log x. y f(x) = log x x Lesson.4 l Transformations of Logarithmic Functions 303

26 3. Sketch and label the graph of f (x) In( x) using the graph of f (x) In x. y f(x) = In(x) x 6 8 Problem 3 Dilations Earlier you learned that the graph of a function f (x) is dilated a units vertically if the function is multiplied by a constant a: af (x). 1. Sketch and label the graphs of f (x) log x, f (x) log x, and f (x) 1 log x. 304 Chapter l Logarithmic Functions

27 The graph of a function f (x) is dilated horizontally c units if the argument of the function is multiplied by the constant c: f (cx).. Sketch and label the graphs of f (x) log x, f (x) log 3x, and f (x) log 1 3 x. Problem 4 Putting It All Together! 1. Sketch and label the graph of f (x) In x 3 using the graph of f (x) In x. y f(x) = In x x Lesson.4 l Transformations of Logarithmic Functions 305

28 . Sketch and label the graph of f (x) In(x ) 3 using the graph of f (x) In x. y 4 3 f(x) = In x x 3 3. Sketch and label the graph of f (x) log( (x )) using the graph of f (x) log x. y f(x) = log x x 3 Be prepared to share your methods and solutions. 306 Chapter l Logarithmic Functions