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Chapter 3 Exponential and Logarithmic Functions Course Number Instructor Section 3.1 Exponential Functions and Their Graphs Date Objective: In this lesson you learned how to recognize, evaluate, and graph exponential functions. Important Vocabulary Define each term or concept. Transcendental functions Functions that are not algebraic. Natural base e The irrational number e ≈ 2.718281828 . . . I. Exponential Functions (Page 184) What you should learn How to recognize and evaluate exponential functions with base a Polynomial functions and rational functions are examples of algebraic functions. The exponential function f with base a is denoted by f(x) = ax , where a > 0, a ≠1, and x is any real number. Example 1: Use a calculator to evaluate the expression 5 3 / 5 . 2.626527804 II. Graphs of Exponential Functions (Pages 185−187) What you should learn How to graph exponential functions with base a For a > 1, is the graph of f ( x) = a x increasing or decreasing over its domain? Increasing For a > 1, is the graph of g ( x) = a − x increasing or decreasing over its domain? Decreasing For the graph of y = a x or y = a − x , a > 1, the domain is (− ∞, ∞) the intercept is the x-axis , the range is (0, 1) (0, ∞) 5 , and 3 . Also, both graphs have as a horizontal asymptote. Example 2: Sketch the graph of the function f ( x) = 3 − x . y 1 -5 -3 -1 -1 1 3 5 x -3 -5 Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved. 43 44 Chapter 3 Exponential and Logarithmic Functions III. The Natural Base e (Pages 187−189) What you should learn How to recognize, evaluate, and graph exponential functions with base e The natural exponential function is given by the function f(x) = ex . Example 3: Use a calculator to evaluate the expression e 3 / 5 . 1.8221188 For the graph of f ( x) = e x , the domain is the range is (0, ∞) (− ∞, ∞) , and the intercept is (0, 1) , . The number e can be approximated by the expression (1 + 1/x)x for large values of x. IV. Applications (Pages 190−192) After t years, the balance A in an account with principal P and annual interest rate r (in decimal form) is given by the formulas: For n compoundings per year: A = P(1 + r/n)nt For continuous compounding: A = Pert What you should learn How to use exponential functions to model and solve real-life problems Example 4: Find the amount in an account after 10 years if $6000 is invested at an interest rate of 7%, (a) compounded monthly. (b) compounded continuously. (a) $12,057.97 (b) $12,082.52 Homework Assignment Page(s) Exercises Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved. Section 3.2 45 Logarithmic Functions and Their Graphs Course Number Section 3.2 Logarithmic Functions and Their Graphs Instructor Objective: In this lesson you learned how to recognize, evaluate, and graph logarithmic functions. Important Vocabulary Date Define each term or concept. Common logarithmic function The logarithmic function with base 10. Natural logarithmic function The logarithmic function with base e given by f(x) = ln x, x > 0. I. Logarithmic Functions (Pages 196−197) The logarithmic function with base a is the inverse of the exponential function f ( x) = a x . function What you should learn How to recognize and evaluate logarithmic functions with base a The logarithmic function with base a is defined as f(x) = loga x , for x > 0, a > 0, and a ≠ 1, if and only if x = ay. The notation “ log a x ” is read as “ log .” base a of x The equation x = ay in exponential form is equivalent to the equation y = loga x in logarithmic form. When evaluating logarithms, remember that a logarithm is a(n) exponent . This means that log a x is the to which a must be raised to obtain x exponent . Example 1: Use the definition of logarithmic function to evaluate log 5 125 . 3 Example 2: Use a calculator to evaluate log10 300 . 2.477121255 Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved. 46 Chapter 3 Exponential and Logarithmic Functions Complete the following properties of logarithms: 1) log a 1 = 0 3) log a a x = x 2) log a a = 1 a log a x = x and 4) If log a x = log a y , then x=y . Example 3: Solve the equation log 7 x = 1 for x. x=7 II. Graphs of Logarithmic Functions (Pages 198−199) For a > 1, is the graph of f ( x) = log a x increasing or decreasing over its domain? What you should learn How to graph logarithmic functions with base a Increasing For the graph of f ( x) = log a x , a > 1, the domain is (0, ∞) , the range is the intercept is , and . (1, 0) Also, the graph has (− ∞, ∞) as a vertical the y-axis asymptote. The graph of f ( x) = log a x is a reflection of the graph of f ( x) = a x in the line y = x . Example 4: Sketch the graph of the function f ( x) = log 3 x . 5 y 3 1 -5 -3 -1 -1 1 3 5 x -3 -5 Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved. Section 3.2 III. The Natural Logarithmic Function (Pages 200−202) Complete the following properties of natural logarithms: 1) ln 1 = 3) ln e x = 47 Logarithmic Functions and Their Graphs 0 x 4) If ln x = ln y , then 2) ln e = and x=y e ln x = 1 What you should learn How to recognize, evaluate, and graph natural logarithmic functions x . Example 5: Use a calculator to evaluate ln 10 . 2.302585093 Example 6: Find the domain of the function f ( x) = ln( x + 3) . (−3, ∞) IV. Applications of Logarithmic Functions (Page 202) Describe a real-life situation in which logarithms are used. Answers will vary. What you should learn How to use logarithmic functions to model and solve real-life problems Example 7: A principal P, invested at 6% interest and compounded continuously, increases to an amount K times the original principal after t years, where t ln K . How long will it take the is given by t = 0.06 original investment to double in value? To triple in value? 11.55 years; 18.31 years Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved. 48 Chapter 3 Exponential and Logarithmic Functions Additional notes y y x y y x y x x y x x Homework Assignment Page(s) Exercises Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved. Section 3.3 49 Properties of Logarithms Course Number Section 3.3 Properties of Logarithms Objective: In this lesson you learned how to rewrite logarithmic functions with different bases and how to use properties of logarithms to evaluate, rewrite, expand, or condense logarithmic expressions. I. Change of Base (Page 207) Let a, b, and x be positive real numbers such that a ≠ 1 and b ≠ 1. The change-of-base formula states that . . . loga x can be converted to a different base using any of the following formulas: Base b: loga x = (logb x)/(logb a) Base 10: loga x = (log10 x)/(log10 a) Base e: loga x = (ln x)/(ln a) Instructor Date What you should learn How to rewrite logarithms with different bases Explain how to use a calculator to evaluate log 8 20 . Using the change-of-base formula, evaluate either (log 20) ÷ (log 8) or (ln 20) ÷ (ln 8). The results will be the same: 1.4406 II. Properties of Logarithms (Page 208) Let a be a positive number such that a ≠ 1; let n be a real number; and let u and v be positive real numbers. Complete the following properties of logarithms: 1. log a (uv) = 2. log a u = v 3. log a u n = What you should learn How to use properties of logarithms to evaluate or rewrite logarithmic expressions loga u + loga v loga u − loga v n loga u III. Rewriting Logarithmic Expressions (Page 209) To expand a logarithmic expression means to . . . . use the properties of logarithms to rewrite complicated products, quotients, and exponential forms into simpler sums, differences, What you should learn How to use properties of logarithms to expand or condense logarithmic expressions and products. Example 1: Expand the logarithmic expression ln ln x + 4 ln y − ln 2 xy 4 . 2 Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved. 50 Chapter 3 To condense a logarithmic expression means to . . . . Exponential and Logarithmic Functions use the properties of logarithms to rewrite the expression as the logarithm of a single quantity. Example 2: Condense the logarithmic expression 3 log x + 4 log( x − 1) . log[x3(x − 1)4] IV. Applications of Properties of Logarithms (Page 210) One way of finding a model for a set of nonlinear data is to take the natural log of each of the x-values and y-values of the data set. If the points are graphed and fall on a straight line, then the x-values and the y-values are related by the equation: ln y = m ln x What you should learn How to use logarithmic functions to model and solve real-life problems , where m is the slope of the straight line. Example 3: Find a natural logarithmic equation for the following data that expresses y as a function of x. x 2.718 7.389 20.086 54.598 y 7.389 54.598 403.429 2980.958 ln y = 2 ln x or ln y = ln x2 y y x y x x Homework Assignment Page(s) Exercises Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved. Section 3.4 51 Solving Exponential and Logarithmic Equations Section 3.4 Solving Exponential and Logarithmic Equations Objective: In this lesson you learned how to solve exponential and logarithmic equations. I. Introduction (Page 214) State the One-to-One Property for exponential equations. ax = ay if and only if x = y Course Number Instructor Date What you should learn How to solve simple exponential and logarithmic equations State the One-to-One Property for logarithmic equations. loga x = loga y if and only if x = y State the Inverse Properties for exponential equations and for logarithmic equations. aloga x = x and loga ax = x Describe some strategies for using the One-to-One Properties and the Inverse Properties to solve exponential and logarithmic equations. • • • Rewrite the original equation in a form that allows the use of the One-to-One Properties of exponential or logarithmic functions. Rewrite an exponential equation in logarithmic form and apply the Inverse Property of logarithmic functions. Rewrite a logarithmic equation in exponential form and apply the Inverse Property of exponential functions. 1 for x. 3 (b) Solve 5 x = 0.04 for x. (a) x = 2 (b) x = − 2 Example 1: (a) Solve log 8 x = II. Solving Exponential Equations (Pages 215−216) Describe how to solve the exponential equation 10 x = 90 algebraically. Take the common logarithm of each side of the equation and then use the Inverse Property to obtain: x = log 90. Then use a calculator to approximate the solution by evaluating log 90 ≈ 1.954. What you should learn How to solve more complicated exponential equations Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved. 52 Chapter 3 Exponential and Logarithmic Functions Example 2: Solve e x − 2 − 7 = 59 for x. Round to three decimal places. x ≈ 6.190 III. Solving Logarithmic Equations (Pages 217−219) Describe how to solve the logarithmic equation log 6 ( 4 x − 7) = log 6 (8 − x) algebraically. What you should learn How to solve more complicated logarithmic equations Use the One-to-One Property for logarithms to write the arguments of each logarithm as equal: (4x − 7) = (8 − x). Then solve this resulting linear equation by adding 7 to each side, adding x to each side, and then finally dividing both sides by 5. The solution is x = 3. Example 3: Solve 4 ln 5 x = 28 for x. Round to three decimal places. x ≈ 219.327 Describe a method that can be used to approximate the solutions of an exponential or logarithmic equation using a graphing utility. Use a graphing utility to graph the left side of the equation as y1 and the right side of the equation as y2. Use the intersect feature or the zoom and trace features to approximate the intersection point. IV. Applications of Solving Exponential and Logarithmic Equations (Page 220) Example 4: Use the formula for continuous compounding, A = Pe rt , to find how long it will take $1500 to triple in value if it is invested at 12% interest, compounded continuously. t ≈ 9.155 years What you should learn How to use exponential and logarithmic equations to model and solve reallife problems Homework Assignment Page(s) Exercises Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved. Section 3.5 53 Exponential and Logarithmic Models Section 3.5 Exponential and Logarithmic Models Objective: In this lesson you learned how to use exponential growth models, exponential decay models, Gaussian models, logistic models, and logarithmic models to solve real-life problems. Important Vocabulary Course Number Instructor Date Define each term or concept. Bell-shaped curve The graph of a Gaussian model. Logistic curve A model for describing populations initially having rapid growth followed by a declining rate of growth. Sigmoidal curve Another name for a logistic growth curve. I. Introduction (Page 225) y = aebx, b > 0 The exponential growth model is y = ae–bx, b > 0 The exponential decay model is −(x−b)2 /c The Gaussian model is y = ae The logistic growth model is Logarithmic models are y = a + b log10 x . . y = a/(1 + be−rx) y = a + b ln x . What you should learn How to recognize the five most common types of models involving exponential or logarithmic functions . and . II. Exponential Growth and Decay (Pages 226−228) Example 1: Suppose a population is growing according to the model P = 800e 0.05t , where t is given in years. (a) What is the initial size of the population? (b) How long will it take this population to double? (a) 800 (b) 13.86 years What you should learn How to use exponential growth and decay functions to model and solve real-life problems To estimate the age of dead organic matter, scientists use the carbon dating model R = 1/1012 e−t/8245 , which denotes the ratio R of carbon 14 to carbon 12 present at any time t (in years). Example 2: The ratio of carbon 14 to carbon 12 in a fossil is R = 10−16. Find the age of the fossil. Approximately 75,737 years old Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved. 54 Chapter 3 Exponential and Logarithmic Functions III. Gaussian Models (Page 229) The Gaussian model is commonly used in probability and statistics to represent populations that are distributed normally . What you should learn How to use Gaussian functions to model and solve real-life problems y On a bell-shaped curve, the average value for a population is where the maximum y-value of the function occurs. x Example 3: Draw the basic form of the graph of a Gaussian model. IV. Logistic Growth Models (Page 230) Give an example of a real-life situation that is modeled by a logistic growth model. Answers will vary. One possibility is a bacteria culture that is initially allowed to grow under ideal conditions, and then under less favorable conditions that inhibit growth. What you should learn How to use logistic growth functions to model and solve real-life problems y Example 4: Draw the basic form of the graph of a logistic growth model. V. Logarithmic Models (Page 231) Example 5: The number of kitchen widgets y (in millions) demanded each year is given by the model y = 2 + 3 ln( x + 1) , where x = 0 represents the year 2000 and x ≥ 0. Find the year in which the number of kitchen widgets demanded will be 8.6 million. In 2008 x What you should learn How to use logarithmic functions to model and solve real-life problems Homework Assignment Page(s) Exercises Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved. Section 3.6 55 Nonlinear Models Course Number Section 3.6 Nonlinear Models Objective: In this lesson you learned how to fit exponential, logarithmic, power, and logistic models to sets of data. I. Classifying Scatter Plots (Page 237) When faced with a set of data to be modeled, what is a good first step in selecting which type of model will best fit the data? Instructor Date What you should learn How to classify scatter plots Making a scatter plot of the data. II. Fitting Nonlinear Models to Data (Pages 237−239) Describe how to use a graphing utility to fit a nonlinear model to data. Answers will vary. For instance, enter the paired data into a graphing utility and graph the data. Use this scatter plot to decide what type of model would fit the data best. Then use the regression feature of the graphing utility to find the appropriate model, either quadratic, exponential, power, or logarithmic. Graph the data and the model in the same viewing window to see whether the model is a good fit to the data. If deciding among several models, compare the coefficients of determination for each model. The model whose r2-value is closest to 1 is the model that best fits the data. What you should learn How to use scatter plots and a graphing utility to find models for data and choose a model that best fits a set of data Example 2: Find an appropriate model, either logarithmic or exponential, for the data in the following table. x y 1 1.120 3 2.195 y = 0.8(1.4)x or 5 4.303 7 8.433 9 16.529 y = 0.8e0.336x Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved. 56 Chapter 3 Exponential and Logarithmic Functions III. Modeling With Exponential and Logistic Functions (Pages 240−241) Example 3: Find a logistic model for the data in the following table. 0 5 x y 10 27 y= 15 50 20 73 25 88 What you should learn How to use a graphing utility to find exponential and logistic models for data 30 95 99.88 1 + 19.67e−0.199x Additional notes y y x y x x Homework Assignment Page(s) Exercises Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved.