Chapter 3 Exponential And Logarithmic Functions
Name DateChapter 3Exponential and Logarithmic FunctionsSection 3.1 Exponential Functions and Their GraphsObjective: In this lesson you learned how to recognize, evaluate, andgraph exponential functions.Define each term or concept.Important VocabularyTranscendental functions Functions that are not algebraic.Natural base e The irrational number e2.718281828 . . .I. Exponential Functions (Page 180)What you should learnHow to recognize andevaluate exponentialfunctions with base aPolynomial functions and rational functions are examples ofalgebraicfunctions.The exponential function f with base a is denoted byf(x) ax, where a 0, a 1, and x is any realnumber.Example 1: Use a calculator to evaluate the expression 5 3 / 5 .2.626527804II. Graphs of Exponential Functions (Pages 181 183)What you should learnHow to graphexponential functionswith base aFor a 1, is the graph of f ( x) a x increasing or decreasingover its domain?IncreasingFor a 1, is the graph of g ( x) aover its domain?For the graph of y(, )the intercept isthe x-axisxincreasing or decreasingDecreasinga x or yax, a 1, the domain is, the range is(0, 1)5(0, ), and3. Also, both graphs have1as a horizontal asymptote.Example 2: Sketch the graph of the function f ( x) 3-5xy-3.-1-1135x-3-5Note Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sixth Edition IAECopyright Cengage Learning. All rights reserved.41
42Chapter 3Exponential and Logarithmic FunctionsIII. The Natural Base e (Pages 184 185)What you should learnHow to recognize,evaluate, and graphexponential functionswith base eThe natural exponential function is given by the functionf(x) ex.Example 3: Use a calculator to evaluate the expression e 3 / 5 .1.8221188For the graph of f ( x) e x , the domain isthe range is(0, )(, ), and the intercept is(0, 1),.The number e can be approximated by the expression(1 1/x)xfor large values of x.IV. Applications (Pages 186 188)After t years, the balance A in an account with principal P andannual interest rate r (in decimal form) is given by the formulas:For n compoundings per year:A P(1 r/n)ntFor continuous compounding:A PertWhat you should learnHow to use exponentialfunctions to model andsolve real-life problemsExample 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) 12,082.52Homework AssignmentPage(s)ExercisesNote Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sixth Edition IAECopyright Cengage Learning. All rights reserved.
Section 3.2Logarithmic Functions and Their Graphs43Name DateSection 3.2 Logarithmic Functions and Their GraphsObjective: In this lesson you learned how to recognize, evaluate, andgraph logarithmic functions.Define each term or concept.Important VocabularyCommon logarithmic function The logarithmic function with base 10.Natural logarithmic function The logarithmic function with base e given byf(x) ln x, x 0.I. Logarithmic Functions (Pages 192 193)The logarithmic function with base a is theinverseof the exponential function f ( x) a x .functionWhat you should learnHow to recognize andevaluate logarithmicfunctions with base aThe logarithmic function with base a is defined asf(x) loga x, for x 0, a 0, and a1, if andonly if x ay. The notation “ log a x ” is read as “base a of xlog.”The equation x ay in exponential form is equivalent to theequationy loga xin logarithmic form.When evaluating logarithms, remember that a logarithm is a(n)exponent. This means that log a x is theto which a must be raised to obtainxexponent.Example 1: Use the definition of logarithmic function toevaluate log 5 125 .3Example 2: Use a calculator to evaluate log 10 300 .2.477121255Note Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sixth Edition IAECopyright Cengage Learning. All rights reserved.
44Chapter 3Exponential and Logarithmic FunctionsComplete the following properties of logarithms:1) log a 1 03) log a a x 2) log a a a loga x andx4) If log a x log a y , then1x yx.Example 3: Solve the equation log 7 x 1 for x.x 7II. Graphs of Logarithmic Functions (Pages 194 195)For a 1, is the graph of f ( x) log a x increasing or decreasingover its domain?What you should learnHow to graph logarithmicfunctions with base aIncreasingFor the graph of f ( x) log a x , a 1, the domain is(0, ), the range isthe intercept is(1, 0)Also, the graph has(, ), and.the y-axisas a verticalasymptote. The graph of f ( x) log a x is a reflection of thegraph of f ( x) a x inthe line y x.Example 4: Sketch the graph of the function f ( x) log 3 x .5y31-5-3-1-1135x-3-5Note Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sixth Edition IAECopyright Cengage Learning. All rights reserved.
Section 3.2Logarithmic Functions and Their GraphsIII. The Natural Logarithmic Function (Pages 196 197)Complete the following properties of natural logarithms:1) ln 1 3) ln e x 0x4) If ln x ln y , then2) ln e andx ye ln x 1What you should learnHow to recognize,evaluate, and graphnatural logarithmicfunctionsx.Example 5: Use a calculator to evaluate ln 10 .2.302585093Example 6: Find the domain of the function f ( x) ln( x 3) .( 3, )IV. Applications of Logarithmic Functions (Page 198)Describe a real-life situation in which logarithms are used.Answers will vary.What you should learnHow to use logarithmicfunctions to model andsolve real-life problemsExample 7: A principal P, invested at 6% interest andcompounded continuously, increases to an amountK times the original principal after t years, where tln Kis given by t. How long will it take the0.06original investment to double in value? To triple invalue?11.55 years; 18.31 yearsNote Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sixth Edition IAECopyright Cengage Learning. All rights reserved.45
46Chapter 3Exponential and Logarithmic FunctionsAdditional notesyyxyyxyxxyxxHomework AssignmentPage(s)ExercisesNote Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sixth Edition IAECopyright Cengage Learning. All rights reserved.
Section 3.3Properties of Logarithms47Name DateSection 3.3 Properties of LogarithmsObjective: In this lesson you learned how to rewrite logarithmic functions withdifferent bases and how to use properties of logarithms to evaluate,rewrite, expand, or condense logarithmic expressions.I. Change of Base (Page 203)Let a, b, and x be positive real numbers such that a 1 and bThe change-of-base formula states that:loga x can be converted to a different base using any of thefollowing 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)1.What you should learnHow to rewritelogarithms with differentbasesExplain 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:1.4406II. Properties of Logarithms (Page 204)Let a be a positive number such that a 1; let n be a realnumber; and let u and v be positive real numbers. Complete thefollowing properties of logarithms:1. log a (uv) 2. log au v3. log a u n What you should learnHow to use properties oflogarithms to evaluate orrewrite logarithmicexpressionsloga u loga vloga uloga vn loga uIII. Rewriting Logarithmic Expressions (Page 205)To expand a logarithmic expression means tousethe properties of logarithms to rewrite complicated products,quotients, and exponential forms into simpler sums, differences,and productsWhat you should learnHow to use properties oflogarithms to expand orcondense logarithmicexpressions.Example 1: Expand the logarithmic expression lnln x 4 ln yxy 4.2ln 2Note Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sixth Edition IAECopyright Cengage Learning. All rights reserved.
48Chapter 3Exponential and Logarithmic FunctionsTo condense a logarithmic expression means touse theproperties of logarithms to rewrite the expression as thelogarithm of a single quantity.Example 2: Condense the logarithmic expression3 log x 4 log( x 1) .log[x3(x 1)4]IV. Applications of Properties of Logarithms (Page 206)One way of finding a model for a set of nonlinear data is to takethe natural log of each of the x-values and y-values of the dataset. If the points are graphed and fall on a straight line, then thex-values and the y-values are related by the equation:ln y m ln xWhat you should learnHow to use logarithmicfunctions to model andsolve real-life problems, where m is the slope of thestraight line.Example 3: Find a natural logarithmic equation for thefollowing data that expresses y as a function of 58ln y 2 ln xor ln y ln x2yyxyxxHomework AssignmentPage(s)ExercisesNote Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sixth Edition IAECopyright Cengage Learning. All rights reserved.
Section 3.4Solving Exponential and Logarithmic Equations49Name DateSection 3.4 Solving Exponential and Logarithmic EquationsObjective: In this lesson you learned how to solve exponential andlogarithmic equations.I. Introduction (Page 210)State the One-to-One Property for exponential equations.ax ay if and only if x yWhat you should learnHow to solve simpleexponential andlogarithmic equationsState the One-to-One Property for logarithmic equations.loga x loga y if and only if x yState the Inverse Properties for exponential equations and forlogarithmic equations.aloga x xandloga ax xDescribe some strategies for using the One-to-One Propertiesand the Inverse Properties to solve exponential and logarithmicequations.Rewrite the original equation in a form that allows theuse of the One-to-One Properties of exponential orlogarithmic functions.Rewrite an exponential equation in logarithmic form andapply the Inverse Property of logarithmic functions.Rewrite a logarithmic equation in exponential form andapply the Inverse Property of exponential functions.1for x.3(b) Solve 5 x 0.04 for x.(a) x 2 (b) x 2Example 1: (a) Solve log 8 xII. Solving Exponential Equations (Pages 211 212)Describe how to solve the exponential equation 10 x 90algebraically.Take the common logarithm of each side of the equation andthen use the Inverse Property to obtain: x log 90. Then use acalculator to approximate the solution by evaluating log 901.954.What you should learnHow to solve morecomplicated exponentialequationsNote Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sixth Edition IAECopyright Cengage Learning. All rights reserved.
50Chapter 3Exponential and Logarithmic FunctionsExample 2: Solve e x 2places.x 6.1907 59 for x. Round to three decimalIII. Solving Logarithmic Equations (Pages 213 215)Describe how to solve the logarithmic equationlog 6 (4 x 7) log 6 (8 x) algebraically.What you should learnHow to solve morecomplicated logarithmicequationsUse the One-to-One Property for logarithms to write thearguments of each logarithm as equal: (4x 7) (8 x). Thensolve 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 5x 28 for x. Round to three decimalplaces.x 219.327Describe a method that can be used to approximate the solutionsof an exponential or logarithmic equation using a graphingutility.Use a graphing utility to graph the left side of the equation as y1and the right side of the equation as y2. Use the intersect featureor the zoom and trace features to approximate the intersectionpoint.IV. Applications of Solving Exponential and LogarithmicEquations (Page 216)Example 4: Use the formula for continuous compounding,A Pe rt , to find how long it will take 1500 totriple in value if it is invested at 12% interest,compounded continuously.t 9.155 yearsWhat you should learnHow to use exponentialand logarithmic equationsto model and solve reallife problemsHomework AssignmentPage(s)ExercisesNote Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sixth Edition IAECopyright Cengage Learning. All rights reserved.
Section 3.5Exponential and Logarithmic Models51Name DateSection 3.5 Exponential and Logarithmic ModelsObjective: In this lesson you learned how to use exponential growthmodels, exponential decay models, Gaussian models, logisticmodels, and logarithmic models to solve real-life problems.Define each term or concept.Important VocabularyBell-shaped curve The graph of a Gaussian model.Logistic curve A model for describing populations initially having rapid growthfollowed by a declining rate of growth.Sigmoidal curve Another name for a logistic growth curve.I. Introduction (Page 221)y aebx, b 0The exponential growth model is–bxThe exponential decay model isThe Gaussian model isy aeLogarithmic models are(x b)2 /cy a b ln x.y a/(1 be rx)The logistic growth model isy a b log10 xy ae , b 0.What you should learnHow to recognize the fivemost common types ofmodels involvingexponential orlogarithmic functions.and.II. Exponential Growth and Decay (Pages 222 224)Example 1: Suppose a population is growing according to themodel P 800 e 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 todouble?(a) 800(b) 13.86 yearsWhat you should learnHow to use exponentialgrowth and decayfunctions to model andsolve real-life problemsTo estimate the age of dead organic matter, scientists use thecarbon dating model R 1/1012 et/8245, whichdenotes the ratio R of carbon 14 to carbon 12 present at any timet (in years).Example 2: The ratio of carbon 14 to carbon 12 in a fossil isR 10 16. Find the age of the fossil.Approximately 75,737 years oldNote Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sixth Edition IAECopyright Cengage Learning. All rights reserved.
52Chapter 3Exponential and Logarithmic FunctionsIII. Gaussian Models (Page 225)The Gaussian model is commonly used in probability andstatistics to represent populations that aredistributednormally.What you should learnHow to use Gaussianfunctions to model andsolve real-life problemsyOn a bell-shaped curve, the average value for a population iswhere themaximum y-valueof the function occurs.xExample 3: Draw the basic form of the graph of a Gaussianmodel.IV. Logistic Growth Models (Page 226)Give an example of a real-life situation that is modeled by alogistic growth model.Answers will vary. One possibility is a bacteria culture that isinitially allowed to grow under ideal conditions, and then underless favorable conditions that inhibit growth.What you should learnHow to use logisticgrowth functions tomodel and solve real-lifeproblemsyExample 4: Draw the basic form of the graph of a logisticgrowth model.V. Logarithmic Models (Page 227)Example 5: The number of kitchen widgets y (in millions)demanded each year is given by the modely 2 3 ln( x 1) , where x 0 represents the year2000 and x 0. Find the year in which the numberof kitchen widgets demanded will be 8.6 million.In 2008xWhat you should learnHow to use logarithmicfunctions to model andsolve real-life problemsHomework AssignmentPage(s)ExercisesNote Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sixth Edition IAECopyright Cengage Learning. All rights reserved.
Section 3.6Nonlinear Models53Name DateSection 3.6 Nonlinear ModelsObjective: In this lesson you learned how to fit exponential,logarithmic, power, and logistic models to sets of data.I. Classifying Scatter Plots (Page 233)When faced with a set of data to be modeled, what is a good firststep in selecting which type of model will best fit the data?What you should learnHow to classify scatterplotsMaking a scatter plot of the data.II. Fitting Nonlinear Models to Data (Pages 234 235)Describe how to use a graphing utility to fit a nonlinear model todata.Answers will vary. For instance, enter the paired data into agraphing utility and graph the data. Use this scatter plot to decidewhat type of model would fit the data best. Then use theregression feature of the graphing utility to find the appropriatemodel, either quadratic, exponential, power, or logarithmic.Graph the data and the model in the same viewing window to seewhether the model is a good fit to the data. If deciding amongseveral models, compare the coefficients of determination foreach model. The model whose r2-value is closest to 1 is themodel that best fits the data.What you should learnHow to use scatter plotsand a graphing utility tofind models for data andchoose the model thatbest fits a set of dataExample 2: Find an appropriate model, either logarithmic orexponential, for the data in the following table.xy11.12032.195y 0.8(1.4)x54.303or78.433916.529y 0.8e0.336xNote Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sixth Edition IAECopyright Cengage Learning. All rights reserved.
54Chapter 3Exponential and Logarithmic FunctionsIII. Modeling With Exponential and Logistic Functions(Pages 236 237)Example 3: Find a logistic model for the data in the followingtable.xy051027y 155099.881 19.67e20732588What you should learnHow to use a graphingutility to find exponentialand logistic models fordata30950.199xAdditional notesyyxyxxHomework AssignmentPage(s)ExercisesNote Taking Guide for Larson Precalculus with Limits: A Graphing Approach, Sixth Edition IAECopyright Cengage Learning. All rights reserved.
Rewrite a logarithmic equation in exponential form and apply the Inverse Property of exponential functions. Example 1: (a) Solve 3 1 log 8 x for x. (b) Solve 5x 0.04 for x. (a) x 2 (b) x 2 II. Solving Exponential Equations (Pages 211 212) Describe how to solve the exponential equation 10 x 9
324 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions 5.1 The Natural Logarithmic Function: Differentiation DEFINITION OF THE NATURAL LOGARITHMIC FUNCTION The natural logarithmic functionis defined by The domain of the natural logarithmic function is the set of all posi
Chapter 7 — Exponential and logarithmic functions . Exponential and logarithmic functions MB11 Qld-7 97 4 a 83 32 45 2 15 2 93 . MB11 Qld-7 98 Exponential and logarithmic functions c 53 152 32 53 (5
rational functions. O In Chapter 3, you will: Evaluate, analyze, and graph exponential and logarithmic functions. Apply properties of logarithms. Solve exponential and logarithmic equations. Model data using exponential, logarithmic, and logistic functions. O ENDANGERED SPECIES
6.3 Logarithmic Functions 6.7 Exponential and Logarithmic Models 6.4 Graphs of Logarithmic Functions 6.8 Fitting Exponential Models to Data Introduction . Solution When an amount grows at a !xed percent per unit time, the growth is exponential. To !nd A 0 we use the fact that A 0 is the amount at time zero, so A 0 10. To !nd k, use the fact .
Chapter 3 Exponential & Logarithmic Functions Section 3.1 Exponential Functions & Their Graphs Definition of Exponential Function: The exponential function f with base a is denoted f(x) a x where a 0, a 1 and x is any real number. Example 1: Evaluating Exponential Expressions Use
Notes Chapter 5 (Logarithmic, Exponential, and Other Transcendental Functions) Definition of the Natural Logarithmic Function: The natural logarithmic function is defined by The domain of the natural logarithmic
348 Chapter 7 Exponential and Logarithmic Functions 7.1 Lesson WWhat You Will Learnhat You Will Learn Graph exponential growth and decay functions. Use exponential models to solve real-life problems. Exponential Growth and Decay Functions An exponential function has the form y abx, where
Aug 08, 2017 · Name:_ Chapter 5 Problem Set SECTION 5.3 PROBLEM SET: LOGARITHMS AND LOGARITHMIC FUNCTIONS Rewrite each of these exponential expressions in logarithmic form: 1) 3 4 81 2) 10 5 100,000 3) 5 2 0.04 4) 4 1 0.25 5) 16 1/4 2 6) 9 1/2 3 Rewrite each of these logarithmic expressions in exponential form:
