Exponential and Logarithmic Functions

Transcription

1 Name Date Chapter 3 Eponential and Logarithmic Functions Section 3.1 Eponential Functions and Their Graphs Objective: In this lesson ou learned how to recognize, evaluate, and graph eponential functions. Important Vocabular Define each term or concept. Transcendental functions Functions that are not algebraic. Natural base e The irrational number e I. Eponential Functions (Page 180) Polnomial functions and rational functions are eamples of algebraic functions. How to recognize and evaluate eponential functions with base a The eponential function f with base a is denoted b f() = a, where a > 0, a 1, and is an real number. Eample 1: Use a calculator to evaluate the epression / 5 5. II. Graphs of Eponential Functions (Pages ) For a > 1, is the graph of f( ) a increasing or decreasing over its domain? Increasing For a > 1, is the graph of g ( ) a increasing or decreasing How to graph eponential functions with base a over its domain? Decreasing For the graph of a or a, a > 1, the domain is (, ), the range is (0, ), and the intercept is (0, 1). Also, both graphs have the -ais as a horizontal asmptote Eample 2: Sketch the graph of the function f ( ) Note Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sith Edition IAE Copright Cengage Learning. All rights reserved. 41

2 42 Chapter 3 Eponential and Logarithmic Functions III. The Natural Base e (Pages ) The natural eponential function is given b the function f() = e. Eample 3: Use a calculator to evaluate the epression / 5 e. How to recognize, evaluate, and graph eponential functions with base e For the graph of f ( ) e, the domain is (, ), the range is (0, ), and the intercept is (0, 1). The number e can be approimated b the epression (1 + 1/) for large values of. IV. Applications (Pages ) After t ears, the balance A in an account with principal P and annual interest rate r (in decimal form) is given b the formulas: For n compoundings per ear: A = P(1 + r/n) nt How to use eponential functions to model and solve real-life problems For continuous compounding: A = Pe rt Eample 4: Find the amount in an account after 10 ears if $6000 is invested at an interest rate of 7%, (a) compounded monthl. (b) compounded continuousl. (a) $12, (b) $12, Homework Assignment Page(s) Eercises Note Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sith Edition IAE Copright Cengage Learning. All rights reserved.

3 Section 3.2 Logarithmic Functions and Their Graphs 43 Name Date Section 3.2 Logarithmic Functions and Their Graphs Objective: In this lesson ou learned how to recognize, evaluate, and graph logarithmic functions. Important Vocabular Define each term or concept. Common logarithmic function The logarithmic function with base 10. Natural logarithmic function The logarithmic function with base e given b f() = ln, > 0. I. Logarithmic Functions (Pages ) The logarithmic function with base a is the function inverse of the eponential function f ( ) a. How to recognize and evaluate logarithmic functions with base a The logarithmic function with base a is defined as f() = log a, for > 0, a > 0, and a 1, if and onl if = a. The notation log a is read as log base a of. The equation = a in eponential form is equivalent to the equation = log a in logarithmic form. When evaluating logarithms, remember that a logarithm is a(n) eponent. This means that log a is the eponent to which a must be raised to obtain. Eample 1: Use the definition of logarithmic function to evaluate log Eample 2: Use a calculator to evaluate log Note Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sith Edition IAE Copright Cengage Learning. All rights reserved.

4 44 Chapter 3 Eponential and Logarithmic Functions Complete the following properties of logarithms: 1) log a 1 = 0 2) log a = 1 3) log a a = and a a a log = 4) If log log, then =. a a Eample 3: Solve the equation log 7 1 for. = 7 II. Graphs of Logarithmic Functions (Pages ) For a > 1, is the graph of f( ) log a over its domain? Increasing increasing or decreasing How to graph logarithmic functions with base a For the graph of f( ) log a, a > 1, the domain is (0, ), the range is (, ), and the intercept is (1, 0). Also, the graph has the -ais as a vertical asmptote. The graph of f( ) log a is a reflection of the graph of f( ) a in the line =. Eample 4: Sketch the graph of the function 5 f 3 ( ) log Note Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sith Edition IAE Copright Cengage Learning. All rights reserved.

5 Section 3.2 Logarithmic Functions and Their Graphs 45 III. The Natural Logarithmic Function (Pages ) Complete the following properties of natural logarithms: 1) ln 1 = 0 2) ln e = 1 3) ln e = and 4) If ln ln, then =. e ln = How to recognize, evaluate, and graph natural logarithmic functions Eample 5: Use a calculator to evaluate ln Eample 6: Find the domain of the function f ( ) ln( 3). ( 3, ) IV. Applications of Logarithmic Functions (Page 198) Describe a real-life situation in which logarithms are used. Answers will var. How to use logarithmic functions to model and solve real-life problems Eample 7: A principal P, invested at 6% interest and compounded continuousl, increases to an amount K times the original principal after t ears, where t ln K is given b t. How long will it take the 0.06 original investment to double in value? To triple in value? ears; ears Note Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sith Edition IAE Copright Cengage Learning. All rights reserved.

6 46 Chapter 3 Eponential and Logarithmic Functions Additional notes Homework Assignment Page(s) Eercises Note Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sith Edition IAE Copright Cengage Learning. All rights reserved.

7 Section 3.3 Properties of Logarithms 47 Name Date Section 3.3 Properties of Logarithms Objective: In this lesson ou learned how to rewrite logarithmic functions with different bases and how to use properties of logarithms to evaluate, rewrite, epand, or condense logarithmic epressions. I. Change of Base (Page 203) Let a, b, and be positive real numbers such that a 1 and b 1. The change-of-base formula states that: log a can be converted to a different base using an of the following formulas: Base b: log a = (log b )/(log b a) Base 10: log a = (log 10 )/(log 10 a) Base e: log a = (ln )/(ln a) How to rewrite logarithms with different bases Eplain how to use a calculator to evaluate log Using the change-of-base formula, evaluate either (log 20) (log 8) or (ln 20) (ln 8). The results will be the same: II. Properties of Logarithms (Page 204) 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: How to use properties of logarithms to evaluate or rewrite logarithmic epressions 1. log a ( uv) = log a u + log a v u log a = log a u log a v v n log a u = n log a u III. Rewriting Logarithmic Epressions (Page 205) To epand a logarithmic epression means to use the properties of logarithms to rewrite complicated products, quotients, and eponential forms into simpler sums, differences, and products. How to use properties of logarithms to epand or condense logarithmic epressions Eample 1: Epand the logarithmic epression ln + 4 ln ln 2 4 ln. 2 Note Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sith Edition IAE Copright Cengage Learning. All rights reserved.

8 48 Chapter 3 Eponential and Logarithmic Functions To condense a logarithmic epression means to use the properties of logarithms to rewrite the epression as the logarithm of a single quantit. Eample 2: Condense the logarithmic epression 3log 4log( 1). log[ 3 ( 1) 4 ] IV. Applications of Properties of Logarithms (Page 206) One wa of finding a model for a set of nonlinear data is to take the natural log of each of the -values and -values of the data set. If the points are graphed and fall on a straight line, then the -values and the -values are related b the equation: How to use logarithmic functions to model and solve real-life problems straight line. ln = m ln, where m is the slope of the Eample 3: Find a natural logarithmic equation for the following data that epresses as a function of ln = 2 ln or ln = ln 2 Homework Assignment Page(s) Eercises Note Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sith Edition IAE Copright Cengage Learning. All rights reserved.

9 Section 3.4 Solving Eponential and Logarithmic Equations 49 Name Date Section 3.4 Solving Eponential and Logarithmic Equations Objective: In this lesson ou learned how to solve eponential and logarithmic equations. I. Introduction (Page 210) State the One-to-One Propert for eponential equations. a = a if and onl if = How to solve simple eponential and logarithmic equations State the One-to-One Propert for logarithmic equations. log a = log a if and onl if = State the Inverse Properties for eponential equations and for logarithmic equations. a log a = and log a a = Describe some strategies for using the One-to-One Properties and the Inverse Properties to solve eponential and logarithmic equations. Rewrite the original equation in a form that allows the use of the One-to-One Properties of eponential or logarithmic functions. Rewrite an eponential equation in logarithmic form and appl the Inverse Propert of logarithmic functions. Rewrite a logarithmic equation in eponential form and appl the Inverse Propert of eponential functions. Eample 1: (a) Solve 1 log 8 for. 3 (b) Solve for. (a) = 2 (b) = 2 II. Solving Eponential Equations (Pages ) Describe how to solve the eponential equation algebraicall. Take the common logarithm of each side of the equation and then use the Inverse Propert to obtain: = log 90. Then use a calculator to approimate the solution b evaluating log How to solve more complicated eponential equations Note Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sith Edition IAE Copright Cengage Learning. All rights reserved.

10 50 Chapter 3 Eponential and Logarithmic Functions 2 Eample 2: Solve e 7 59 for. Round to three decimal places III. Solving Logarithmic Equations (Pages ) Describe how to solve the logarithmic equation log 6 (4 7) log 6 (8 ) algebraicall. How to solve more complicated logarithmic equations Use the One-to-One Propert for logarithms to write the arguments of each logarithm as equal: (4 7) = (8 ). Then solve this resulting linear equation b adding 7 to each side, adding to each side, and then finall dividing both sides b 5. The solution is = 3. Eample 3: Solve 4 ln 5 28 for. Round to three decimal places Describe a method that can be used to approimate the solutions of an eponential or logarithmic equation using a graphing utilit. Use a graphing utilit to graph the left side of the equation as 1 and the right side of the equation as 2. Use the intersect feature or the zoom and trace features to approimate the intersection point. IV. Applications of Solving Eponential and Logarithmic Equations (Page 216) Eample 4: Use the formula for continuous compounding, rt A Pe, to find how long it will take $1500 to triple in value if it is invested at 12% interest, compounded continuousl. t ears How to use eponential and logarithmic equations to model and solve reallife problems Homework Assignment Page(s) Eercises Note Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sith Edition IAE Copright Cengage Learning. All rights reserved.

11 Section 3.5 Eponential and Logarithmic Models 51 Name Date Section 3.5 Eponential and Logarithmic Models Objective: In this lesson ou learned how to use eponential growth models, eponential deca models, Gaussian models, logistic models, and logarithmic models to solve real-life problems. Important Vocabular Define each term or concept. Bell-shaped curve The graph of a Gaussian model. Logistic curve A model for describing populations initiall having rapid growth followed b a declining rate of growth. Sigmoidal curve Another name for a logistic growth curve. I. Introduction (Page 221) The eponential growth model is = ae b, b > 0. The eponential deca model is = ae b, b > 0. The Gaussian model is = ae ( b) /c. The logistic growth model is = a/(1 + be r ). Logarithmic models are = a + b ln and = a + b log How to recognize the five most common tpes of models involving eponential or logarithmic functions II. Eponential Growth and Deca (Pages ) Eample 1: Suppose a population is growing according to the 0.05t model P 800 e, where t is given in ears. (a) What is the initial size of the population? (b) How long will it take this population to double? (a) 800 (b) ears How to use eponential growth and deca functions to model and solve real-life problems To estimate the age of dead organic matter, scientists use the carbon dating model R = 1/10 12 e t/8245, which denotes the ratio R of carbon 14 to carbon 12 present at an time t (in ears). Eample 2: The ratio of carbon 14 to carbon 12 in a fossil is R = Find the age of the fossil. Approimatel 75,737 ears old Note Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sith Edition IAE Copright Cengage Learning. All rights reserved.

12 52 Chapter 3 Eponential and Logarithmic Functions III. Gaussian Models (Page 225) The Gaussian model is commonl used in probabilit and statistics to represent populations that are normall distributed. How to use Gaussian functions to model and solve real-life problems On a bell-shaped curve, the average value for a population is where the maimum -value of the function occurs. Eample 3: Draw the basic form of the graph of a Gaussian model. IV. Logistic Growth Models (Page 226) Give an eample of a real-life situation that is modeled b a logistic growth model. Answers will var. One possibilit is a bacteria culture that is initiall allowed to grow under ideal conditions, and then under less favorable conditions that inhibit growth. How to use logistic growth functions to model and solve real-life problems Eample 4: Draw the basic form of the graph of a logistic growth model. V. Logarithmic Models (Page 227) Eample 5: The number of kitchen widgets (in millions) demanded each ear is given b the model 2 3 ln( 1), where = 0 represents the ear 2000 and 0. Find the ear in which the number of kitchen widgets demanded will be 8.6 million. In 2008 How to use logarithmic functions to model and solve real-life problems Homework Assignment Page(s) Eercises Note Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sith Edition IAE Copright Cengage Learning. All rights reserved.

13 Section 3.6 Nonlinear Models 53 Name Date Section 3.6 Nonlinear Models Objective: In this lesson ou learned how to fit eponential, logarithmic, power, and logistic models to sets of data. I. Classifing Scatter Plots (Page 233) When faced with a set of data to be modeled, what is a good first step in selecting which tpe of model will best fit the data? How to classif scatter plots Making a scatter plot of the data. II. Fitting Nonlinear Models to Data (Pages ) Describe how to use a graphing utilit to fit a nonlinear model to data. Answers will var. For instance, enter the paired data into a graphing utilit and graph the data. Use this scatter plot to decide what tpe of model would fit the data best. Then use the regression feature of the graphing utilit to find the appropriate model, either quadratic, eponential, 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 r 2 -value is closest to 1 is the model that best fits the data. How to use scatter plots and a graphing utilit to find models for data and choose the model that best fits a set of data Eample 2: Find an appropriate model, either logarithmic or eponential, for the data in the following table = 0.8(1.4) or = 0.8e Note Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sith Edition IAE Copright Cengage Learning. All rights reserved.

14 54 Chapter 3 Eponential and Logarithmic Functions III. Modeling With Eponential and Logistic Functions (Pages ) Eample 3: Find a logistic model for the data in the following table How to use a graphing utilit to find eponential and logistic models for data = e Additional notes Homework Assignment Page(s) Eercises Note Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sith Edition IAE Copright Cengage Learning. All rights reserved.