Mathematics
Graphs of Exponents and Logarithms
Graphs of exponents and logarithms show the relationship between the base and the exponent or logarithm. Exponential graphs typically have a curved shape, while logarithmic graphs are characterized by a curve that approaches but never touches the x-axis. These graphs are useful for visualizing and understanding the behavior of exponential and logarithmic functions.
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9 Key excerpts on "Graphs of Exponents and Logarithms"
- Cynthia Y. Young(Author)
- 2016(Publication Date)
- Wiley(Publisher)
Last, we will discuss particular exponential and logarithmic models that represent phenomena such as compound interest, world populations, conservation biology models, carbon dating, pH values in chemistry, and the bell curve that is fundamental in statistics for describing how quantities vary in the real world. 425 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 5.1 EXPONENTIAL FUNCTIONS AND THEIR GRAPHS 5.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS 5.3 PROPERTIES OF LOGARITHMS 5.4 EXPONENTIAL AND LOGARITHMIC EQUATIONS 5.5 EXPONENTIAL AND LOGARITHMIC MODELS • Evaluating Exponential Functions • Graphs of Exponential Functions • The Natural Base e • Applications of Exponential Functions • Evaluating Logarithms • Common and Natural Logarithms • Graphs of Logarithmic Functions • Applications of Logarithms • Properties of Logarithmic Functions • Change-of-Base Formula • Exponential Equations • Solving Logarithmic Equations • Exponential Growth Models • Exponential Decay Models • Gaussian (Normal) Distribution Models • Logistic Growth Models • Logarithmic Models 426 CHAPTER 5 Exponential and Logarithmic Functions 5.1.1 Evaluating Exponential Functions Most of the functions (polynomial, rational, radical, etc.) we have studied thus far have been algebraic functions. Algebraic functions involve basic operations, powers, and roots. In this chapter, we discuss exponential functions and logarithmic functions, which are called transcendental functions because they transcend our ability to define them with a finite number of algebraic expressions. The following table illustrates the difference between algebraic functions and exponential functions. FUNCTION VARIABLE IS IN THE CONSTANT IS IN THE EXAMPLE EXAMPLE Algebraic Base Exponent ƒ1x2 5 x 2 g 1x2 5 x 1/3 Exponential Exponent Base F 1x2 5 2 x G1 x2 5 a 1 3 b x DEFINITION Exponential Function An exponential function with base b is denoted by ƒ 1 x 2 5 b x where b and x are any real numbers such that b .
eBook - PDF- Sheldon Axler(Author)
- 2011(Publication Date)
- Wiley(Publisher)
For example, 2 x > x 1000 for all x > 13747 (Problem 35 shows that 13747 could not be replaced by 13746). Functions with exponential growth increase so rapidly that graphing them in the usual manner can display too little information, as shown in the following example. 330 chapter 5 Exponents and Logarithms example 1 Discuss the graph of the function 9 x on the interval [0, 8]. solution 1 2 3 4 5 6 7 8 x 10 000 000 20 000 000 30 000 000 40 000 000 y The graph of y = 9 x on the interval [0, 8]. The graph of the function 9 x on the interval [0, 8] is shown above. In this graph, we cannot use the same scale on the x- and y -axes because 9 8 is larger than forty million. Due to the scale, the shape of the graph in the interval [0, 5] gives little insight into the behavior of the function there. For example, this graph does not adequately distinguish between the values 9 2 (which equals 81) and 9 5 (which equals 59049). Because the graphs of functions with exponential growth often do not provide sufficient visual information, data that is expected to have expo- nential growth is often graphed by taking the logarithm of the data. The advantage of this procedure is that if f is a function with exponential growth, then the logarithm of f is a linear function. For example, if f (x) = 2 x , then Here we are taking the logarithm base 10, but the conclu- sion about the linear- ity of the logarithm of f would hold re- gardless of the base used for the logarithm. log f (x) = (log 2)x; thus the graph of log f is the line whose equation is y = (log 2)x (which is the line through the origin with slope log 2). More generally, if f (x) = cb kx , then log f (x) = log c + log b kx = k(log b)x + log c. Here k, log b, and log c are all constants; thus the function log f is indeed linear. If k > 0 and b > 1, as is required in the definition of exponential growth, then k log b > 0, which implies that the line y = log f (x) has positive slope.
eBook - PDF- Cynthia Y. Young(Author)
- 2021(Publication Date)
- Wiley(Publisher)
We call this exponent a logarithm (or “log” for short). Words Math x = b y is equivalent to y = log b x. y = log b x Let y = f −1 (x). f −1 (x) = log b x Logarithmic Function For x > 0, b > 0, and b ≠ 1, the logarithmic function with base b is denoted f (x) = log b x, where y = log b x if and only if x = b y We read log b x as “log base b of x.” 5.2 Logarithmic Functions and Their Graphs 441 STUDY TIP log b x = y is equivalent to b y = x. This definition says that x = b y (exponential form) and y = log b x (logarithmic form) are equivalent. One way to remember this relationship is by adding arrows to the logarithmic form: log b x = y ⇔ b y = x EXAMPLE 1 Changing from Logarithmic Form to Exponential Form Write each equation in its equivalent exponential form. a. log 2 8 = 3 b. log 9 3 = 1 _ 2 c. log 5( 1 __ 25 ) = −2 Solution a. log 2 8 = 3 is equivalent to 2 3 = 8 b. log 9 3 = 1 _ 2 is equivalent to 9 1/2 = 3 c. log 5( 1 __ 25 ) = −2 is equivalent to 5 −2 = 1 __ 25 Your Turn Write each equation in its equivalent exponential form. a. log 3 9 = 2 b. log 16 4 = 1 _ 2 c. log 2( 1 _ 8 ) = −3 Answer a. 3 2 = 9 b. 16 1/2 = 4 c. 2 −3 = 1 _ 8 EXAMPLE 2 Changing from Exponential Form to Logarithmic Form Write each equation in its equivalent logarithmic form. a. 16 = 2 4 b. 9 = √ _ 81 c. 1 _ 9 = 3 −2 d. x a = z Solution a. 16 = 2 4 is equivalent to log 2 16 = 4 b. 9 = √ _ 81 = 81 1/2 is equivalent to log 81 9 = 1 _ 2 c. 1 _ 9 = 3 −2 is equivalent to log 3( 1 _ 9 ) = −2 d. x a = z is equivalent to log x z = a Your Turn Write each equation in its equivalent logarithmic form. a. 81 = 9 2 b. 12 = √ _ 144 c. 1 __ 49 = 7 −2 d. y b = w Answer a. log 9 81 = 2 b. log 144 12 = 1 _ 2 c. log 7 ( 1 _ 49 ) = −2 d. log y w = b 442 CHAPTER 5 Exponential and Logarithmic Functions Some logarithms can be found exactly, whereas others must be approximated. Example 3 illustrates how to find the exact value of a logarithm. Example 4 illustrates approximating values of logarithms with a calculator.
eBook - PDF- Ron Larson(Author)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
Solve more complicated exponential equations. Solve more complicated logarithmic equations. Use exponential and logarithmic equations to model and solve real-life problems. Introduction So far in this chapter, you have studied the definitions, graphs, and properties of exponential and logarithmic functions. In this section, you will study procedures for solving equations involving exponential and logarithmic expressions. There are two basic strategies for solving exponential or logarithmic equations. The first is based on the One-to-One Properties and was used to solve simple exponential and logarithmic equations in Sections 5.1 and 5.2. The second is based on the Inverse Properties. For a > 0 and a ≠ 1, the properties below are true for all x and y for which log a x and log a y are defined. One-to-One Properties Inverse Properties a x = a y if and only if x = y. a log a x = x log a x = log a y if and only if x = y. log a a x = x Solving Simple Equations Original Rewritten Equation Equation Solution Property a. 2 x = 32 2 x = 2 5 x = 5 One-to-One b. ln x - ln 3 = 0 ln x = ln 3 x = 3 One-to-One c. ( 1 3 ) x = 9 3 -x = 3 2 x = -2 One-to-One d. e x = 7 ln e x = ln 7 x = ln 7 Inverse e. ln x = -3 e ln x = e -3 x = e -3 Inverse f. log x = -1 10 log x = 10 -1 x = 10 -1 = 1 10 Inverse g. log 3 x = 4 3 log 3 x = 3 4 x = 81 Inverse Checkpoint Audio-video solution in English & Spanish at LarsonPrecalculus.com Solve each equation for x. a. 2 x = 512 b. log 6 x = 3 c. 5 - e x = 0 d. 9 x = 1 3 Strategies for Solving Exponential and Logarithmic Equations 1. Rewrite the original equation in a form that allows the use of the One-to-One Properties of exponential or logarithmic functions. 2. Rewrite an exponential equation in logarithmic form and apply the Inverse Property of logarithmic functions. 3. Rewrite a logarithmic equation in exponential form and apply the Inverse Property of exponential functions. Exponential and logarithmic equations have many life science applications.
eBook - PDF- R. Gustafson, Jeff Hughes(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Chapter 5 Exponential and Logarithmic Functions 530 Graph: f sxd 5 log 6 x. Now Try Exercise 105. The graphs of all logarithmic functions are similar to those in Figure 5-21. If b . 1, the logarithmic function is an increasing function, as in Figure 5-21(a). If 0 , b , 1, the logarithmic function is a decreasing function, as in Figure 5-21(b). We also know that the exponential function f sxd 5 b x and the logarithmic function f sxd 5 log b x are inverse functions and therefore symmetric about the line y 5 x, as in Figure 5-21(c) and (d). y x (a) (b, 1) (1, 0) f (x) = log b x b > 1 y x (b) (b, 1) (1, 0) f (x) = log b x 0 < b < 1 y x (c) y = b x y = log b x b > 1 y x (d) y = b x y = log b x 0 < b < 1 FIGURE 5-21 To graph, f sxd 5 log 10 x, we can plot points that satisfy the equation x 5 10 y and join them with a smooth curve, as shown in Figure 5-22. The Graph of the Common Logarithm Function f sxd 5 log 10 x Self Check 8 FIGURE 5-22 f (x) = log x x y f sxd 5 logx x f sxd sx, f sxdd 1 100 22 1 1 100 , 22 2 1 10 21 1 1 10 , 21 2 1 0 s1, 0d 10 1 s10, 1d 100 2 s100, 2d Tip The base of the common logarithmic function is 10. Choose x-values that are integer powers of 10. The function f sxd 5 log 10 x is very important and its graph should be memorized. Place in your library of functions. Take Note As Figure 5-22 shows, the graph of f sxd 5 log b x has these properties: Properties of the Graph of f sxd 5 log b x 1. It passes through the point s1, 0d. 2. It passes through the point sb, 1d. 3. The y-axis is a vertical asymptote. 4. The domain is s0, `d, and the range is s2`, `d. To graph f sxd 5 ln x, we can plot points that satisfy the equation x 5 e y and join them with a smooth curve, as shown in Figure 5-23. Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
eBook - PDF- Ron Larson(Author)
- 2021(Publication Date)
- Cengage Learning EMEA(Publisher)
Use the graph of y = log 3 x + log 3 (x - 8) - 2 to determine whether each value is an actual solution of the equation. Explain. x y (9, 0) 3 6 12 15 - 3 3 Copyright 2022 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 398 Chapter 5 Exponential and Logarithmic Functions 1.4 Functions GO DIGITAL 5.5 Exponential and Logarithmic Models Recognize the five most common types of models involving exponential and logarithmic functions. Use exponential growth and decay functions to model and solve real-life problems. Use Gaussian functions to model and solve real-life problems. Use logistic growth functions to model and solve real-life problems. Use logarithmic functions to model and solve real-life problems. Introduction The five most common types of mathematical models involving exponential functions and logarithmic functions are listed below. 1. Exponential growth model: y = ae bx , b > 0 2. Exponential decay model: y = ae -bx , b > 0 3. Gaussian model: y = ae -(x -b) 2 c 4. Logistic growth model: y = a 1 + be -rx 5. Logarithmic models: y = a + b ln x, y = a + b log x The basic shapes of the graphs of these functions are shown below.
eBook - PDF- Cynthia Y. Young(Author)
- 2023(Publication Date)
- Wiley(Publisher)
302 CHAPTER 3 Exponential and Logarithmic Functions All of the transformation techniques (shifting, reflection, and compression) discussed in Chapter 1 also apply to logarithmic functions. For example, the graphs of −log 2 x and log 2 (−x) are found by reflecting the graph of y = log 2 x about the x-axis and y-axis, respectively. x (2, 1) (2, –1) (– 4, 2) (4, 2) (4, –2) (–2, 1) Concept Check Find the x-intercept, domain, and range of log b (x − 1). Answer: x-intercept: (2, 0); Domain: (1, ∞); Range: (−∞, ∞). Video EXAMPLE 8 Graphing Logarithmic Functions Using Transformations Graph the function f (x) = −log 2 (x − 3) and state its domain and range. Solution Graph y = log 2 x. x-intercept: (1, 0) Vertical asymptote: x = 0 Additional points: (2, 1), (4, 2) Graph y = log 2 (x − 3) by shifting y = log 2 x to the right three units. x-intercept: (4, 0) Vertical asymptote: x = 3 Additional points: (5, 1), (7, 2) Graph y = −log 2 (x − 3) by reflecting y = log 2 (x − 3) about the x-axis. x-intercept: (4, 0) Vertical asymptote: x = 3 Additional points: (5, −1), (7, −2) Domain: (3, ∞) Range: (−∞, ∞) x y 10 5 –5 (4, 2) (2, 1) (1, 0) x y 10 5 –5 (7, 2) (5, 1) (4, 0) x y 10 5 –5 (4, 0) (5, –1) (7, –2) 3.2 Logarithmic Functions and Their Graphs 303 3.2.4 Applications of Logarithms 3.2.4 Skill Apply logarithmic functions to problems in the natural sciences and engineering. 3.2.4 Conceptual Understand that logarithmic functions allow very large ranges of numbers in science and engineering applications to be represented on a smaller scale. Logarithms are used to make a large range of numbers manageable. For example, to create a scale to measure a human’s ability to hear, we must have a way to measure the sound intensity of an explosion, even though that intensity can be more than a trillion (10 12 ) times greater than that of a soft whisper. Decibels in engineering and physics, pH in chemistry, and the Richter scale for earthquakes are all applications of logarithmic functions.
eBook - PDF- Ron Larson(Author)
- 2021(Publication Date)
- Cengage Learning EMEA(Publisher)
Use the graph of y = log 3 x + log 3 (x - 8) - 2 to determine whether each value is an actual solution of the equation. Explain. x y (9, 0) 3 6 12 15 - 3 3 Copyright 2022 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 236 Chapter 3 Exponential and Logarithmic Functions 3.5 Exponential and Logarithmic Models GO DIGITAL 3.5 Exponential and Logarithmic Models Recognize the five most common types of models involving exponential and logarithmic functions. Use exponential growth and decay functions to model and solve real-life problems. Use Gaussian functions to model and solve real-life problems. Use logistic growth functions to model and solve real-life problems. Use logarithmic functions to model and solve real-life problems. Introduction The five most common types of mathematical models involving exponential functions and logarithmic functions are listed below. 1. Exponential growth model: y = ae bx , b > 0 2. Exponential decay model: y = ae -bx , b > 0 3. Gaussian model: y = ae -(x -b) 2 c 4. Logistic growth model: y = a 1 + be -rx 5. Logarithmic models: y = a + b ln x, y = a + b log x The basic shapes of the graphs of these functions are shown below.
eBook - PDF- Cynthia Y. Young(Author)
- 2018(Publication Date)
- Wiley(Publisher)
The first step in this process consists of writing y 5 k x in an equivalent form using the natural logarithm. Use the properties of this section to write an equivalent form of the following implicitly defined functions. 103. y 5 2 x 104. y 5 4 x ⋅ 3 x11 3.4 Exponential and Logarithmic Equations 325 326 CHAPTER 3 Exponential and Logarithmic Functions SKILLS OBJECTIVES ■ ■ Use exponential growth functions to model populations, economic applications of assets, and initial spreading of diseases. ■ ■ Use exponential decay functions to model scenarios in medicine, economic depreciation of assets, archaeology, and forensic science. ■ ■ Represent distributions using a Gaussian (normal) model. ■ ■ Represent restricted growth with logistic models. ■ ■ Use logistic growth models to determine time to pay off debt. CONCEPTUAL OBJECTIVES ■ ■ Understand that exponential growth models assume “uninhibited” growth. ■ ■ Understand that the horizontal asymptote corresponds to a limiting value, often zero. ■ ■ Understand that the bell curve is used to represent standardized tests (IQ/SAT), as well as height/weight charts. ■ ■ Understand that the asymptote of a logistic model corresponds to the carrying capacity of the system. ■ ■ Understand why doubling your monthly payment will reduce the life of the loan by more than half. 3.5 EXPONENTIAL AND LOGARITHMIC MODELS NAME MODEL GRAPH APPLICATIONS Exponential growth ƒ1 t 2 5 ce kt k . 0 World populations, bacteria growth, appreciation, global spread of the HIV virus Exponential decay ƒ1 t 2 5 ce 2kt k .
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