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Chapter 2 Functions and Graphs Section 6 Logarithmic Functions (Part I) Learning Objectives for Section 2.6 Logarithmic Functions The student will be able to: • Identify the graphs of one-to-one functions. • Use and apply inverse functions. • Evaluate logarithms. • Rewrite log as exponential functions and vice versa. Barnett/Ziegler/Byleen Business Calculus 12e 2 One to One Functions Definition: A function f is said to be one-to-one if no x or y values are represented more than once. • One-to-one: 𝑓 = { −1, −1 , 0,0 , 1,1 , 2,8 , 3,27 } • Not one-to-one: 𝑔 = { −2,4 , −1,1 , 0,0 , 1,1 , 2,4 } The graph of a one-to-one function passes both the vertical and horizontal line tests. Barnett/Ziegler/Byleen Business Calculus 12e 3 Which Functions Are One to One? 40 12 30 10 8 20 6 10 4 0 -4 -2 0 2 2 4 -10 -20 -30 0 -4 -2 0 2 4 NOT One-to-one One-to-one Barnett/Ziegler/Byleen Business Calculus 12e 4 Definition of Inverse Function If f is a one-to-one function, then the inverse of f is the function formed by interchanging the x and y coordinates for f. Thus, if (a, b) is a point on the graph of f, then (b, a) is a point on the graph of the inverse of f. • Let 𝑓 = 2, 5 , 5, 11 , −1, −1 , 0,1 • Then 𝑓 −1 = { 5,2 , 11,5 , −1, −1 , 1,0 } The domain of f becomes the range of 𝑓 −1 . The range of f becomes the domain of 𝑓 −1 . Note: If a function is not one-to-one then f does not have an inverse. Barnett/Ziegler/Byleen Business Calculus 12e 5 Finding the Inverse Function Given the equation of a one-to-one function f, you can find 𝑓 −1 algebraically by exchanging x for y and solving for y. Example: Find 𝑓 −1 𝑥 = −2y − 3 𝑦 = −2x − 3 𝑥 + 3 = −2y 𝑥+3 =y −2 𝑥+3 −1 𝑓 𝑥 = −2 Barnett/Ziegler/Byleen Business Calculus 12e 6 Graphs of f and f-1 The graphs of 𝑓 and 𝑓 −1 are reflections of each other over the line 𝒚 = 𝒙 If you know how to graph 𝑓 then simply take a few key points and switch their x and y coordinates to help you graph 𝑓 −1 . Or find the equation of 𝑓 −1 algebraically first, then graph it. Barnett/Ziegler/Byleen Business Calculus 12e 7 Graphs of f and f-1 Graph 𝑓 and 𝑓 −1 (from the previous example) on the same coordinate plane. 𝑓 𝑥 = −2x − 3 𝑓 −1 𝑓 −1 𝑥+3 𝑥 = −2 1 3 𝑥 =− 𝑥− 2 2 𝑓 𝑓 −1 Barnett/Ziegler/Byleen Business Calculus 12e 8 Graphs of f and f-1 The graph of 𝑓 is shown. Graph 𝑓 −1 . 𝑦=𝑥 Barnett/Ziegler/Byleen Business Calculus 12e 9 Logarithmic Functions Exponential functions are one-to-one because they pass the vertical and horizontal line tests. 𝑦 = 2𝑥 Barnett/Ziegler/Byleen Business Calculus 12e 10 Inverse of an Exponential Function Start with the exponential function: 𝑦 = 2𝑥 Now, interchange x and y: 𝑥 = 2𝑦 Solving for y: 𝑦 = 𝑙𝑜𝑔2 𝑥 This is called a logarithmic function. The inverse of an exponential function is a log function. 𝑓 𝑥 = 2𝑥 𝑓 −1 𝑥 = 𝑙𝑜𝑔2 𝑥 Barnett/Ziegler/Byleen Business Calculus 12e 11 Logarithmic Function The inverse of an exponential function is called a logarithmic function. For b > 0 and b 1, 𝐸𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙 𝑓(𝑥) = 𝑏 𝑥 𝐿𝑜𝑔𝑎𝑟𝑖𝑡ℎ𝑚𝑖𝑐 𝑓 −1 𝑥 = 𝑙𝑜𝑔𝑏 𝑥 𝐷𝑜𝑚𝑎𝑖𝑛: −∞, ∞ 𝐷𝑜𝑚𝑎𝑖𝑛: 0, ∞ 𝑅𝑎𝑛𝑔𝑒: 0, ∞ 𝑅𝑎𝑛𝑔𝑒: −∞, ∞ Barnett/Ziegler/Byleen Business Calculus 12e 12 Graphs 𝐹𝑜𝑟 𝑦 = 2𝑥 D: −∞, ∞ 𝑅: (0, ∞) D: 0, ∞ 𝑅: −∞, ∞ 𝐹𝑜𝑟 𝑦 = log 2 𝑥 Barnett/Ziegler/Byleen Business Calculus 12e 13 Transformations Parent function: 𝑦 = 𝑙𝑜𝑔𝑏 𝑥 Children: 𝑦 = 𝑙𝑜𝑔𝑏 𝑥 + 2 • Shifted up 2 • Shifted right 5 𝑦 = 𝑙𝑜𝑔𝑏 (𝑥 − 5) • Shifted down 3 and left 7 Barnett/Ziegler/Byleen Business Calculus 12e 𝑦 = 𝑙𝑜𝑔𝑏 𝑥 + 7 − 3 14 Log Notation Common Log • log base 10 • When no base is specified, it’s base 10 • 𝑙𝑜𝑔10 𝑥 log 𝑥 Natural Log • log base e • log 𝑒 𝑥 → ln 𝑥 Barnett/Ziegler/Byleen Business Calculus 12e 15 Simple Logs Evaluate each log expression without a calculator: 10 • 𝑙𝑜𝑔2 210 −3 • 𝑙𝑜𝑔5 5−3 2 • 𝑙𝑜𝑔3 9 1 • log 10 • • • • • log 0.01 ln 𝑒 7 ln 𝑒 log 1 ln 1 −2 7 1 0 0 Barnett/Ziegler/Byleen Business Calculus 12e 16 Log Exponential Think of the word “log” as meaning “exponent on base b” To convert a log equation to an exponential equation: 𝑦 = 𝑙𝑜𝑔3 27 𝑙𝑜𝑔3 27 = 𝑦 • What’s the base? 𝟑 • What’s the exponent? 𝒚 • Write the equation 𝟑𝒚 = 𝟐𝟕 Barnett/Ziegler/Byleen Business Calculus 12e 17 Log Exponential Converting a log into an exponential expression: 1. x = log 4 16 4𝑥 = 16 2. 1 log 𝑥 8 = −3 Barnett/Ziegler/Byleen Business Calculus 12e 𝑥 −3 = 1 8 18 Exponential Log To convert an exponential equation to a log equation: 16 = 2𝑦 • • • • What’s the base? 𝟐 What’s the exponent? 𝒚 Write the equation 𝒚 = 𝒍𝒐𝒈𝟐 𝟏𝟔 Check: 𝒚 = 𝒍𝒐𝒈𝟐 𝟏𝟔 Barnett/Ziegler/Byleen Business Calculus 12e 19 Exponential Log Converting an exponential into a log expression: 1. 53 = 125 3 = log 5 125 2. 10𝑥 = 1 100 Barnett/Ziegler/Byleen Business Calculus 12e 𝑥= 1 log10 100 20 Solving Simple Equations Convert each log to an exponential equation and solve for x: 1. log 𝑥 1000 = 3 𝑥 3 = 1000 3 𝑥 = 1000 𝑥 = 10 2. log 6 𝑥 = 5 65 = 𝑥 𝑥 = 7776 Barnett/Ziegler/Byleen Business Calculus 12e 21 Using Your Calculator Use your calculator to evaluate and round to 2 decimal places: 𝑙𝑛 15 ≈ 2.71 𝑙𝑜𝑔 15 ≈ 1.18 Barnett/Ziegler/Byleen Business Calculus 12e 22 23 Chapter 2 Functions and Graphs Section 6 Logarithmic Functions (Part II) Learning Objectives for Section 2.6 Logarithmic Functions The student will be able to: • Use log properties. • Solve log equations. • Solve exponential equations. Barnett/Ziegler/Byleen Business Calculus 12e 25 Properties of Logarithms If b, M, and N are positive real numbers, b 1, and p and x are real numbers, then 1. log b (1) 0 5. log b MN log b M log b N 2. log b (b) 1 M 6. log b log b M log b N N 7. log b M p p log b M 3. log b b x x 4. b log b x x 8. log b M log b N iff M N log 𝑥 9. 𝐶ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑏𝑎𝑠𝑒 𝑓𝑜𝑟𝑚𝑢𝑙𝑎: log 𝑏 𝑥 = log 𝑏 Barnett/Ziegler/Byleen Business Calculus 12e 26 Using Properties Rewrite each expression by using the appropriate log property: 20 = 𝑙𝑜𝑔2 4 = 2 • 𝑙𝑜𝑔2 20 − 𝑙𝑜𝑔2 5 = 𝑙𝑜𝑔2 5 • 𝑙𝑜𝑔5 25𝑥 = 𝑥𝑙𝑜𝑔5 25 = 2𝑥 • log 10𝑥 = log 10 + log 𝑥 = 1 + log 𝑥 • ln 2𝑥 + 1 = ln(5) 2𝑥 + 1 = 5 𝑥=2 • 7𝑙𝑜𝑔7 (𝑥+1) = 𝑥 + 1 log 19 • 𝑙𝑜𝑔3 19 = ≈ 2.68 log 3 Barnett/Ziegler/Byleen Business Calculus 12e 27 Solving Log Equations Solve for x: log 4 x 6 log 4 x 6 3 log 4 𝑥 + 6 𝑥 − 6 = 3 log 4 𝑥 2 − 36 = 3 43 = 𝑥 2 − 36 64 = 𝑥 2 − 36 x can’t be -10 because you can’t take the log of a negative number. Barnett/Ziegler/Byleen Business Calculus 12e 100 = 𝑥 2 𝑥 = ±10 𝑥 = 10 28 Solving Log Equations Solve for x. Obtain the exact solution of this equation in terms of e. ln (x + 1) – ln x = 1 𝑥+1 𝑙𝑛 =1 𝑥 𝑒1 𝑥+1 = 𝑥 ex - x = 1 x(e - 1) = 1 1 x e 1 ex = x + 1 Barnett/Ziegler/Byleen Business Calculus 12e 29 Solving Exponential Equations Method 1: • Convert the exponential equation to a log equation. • Then evaluate. 9𝑥 = 2 𝑥 = 𝑙𝑜𝑔9 2 log 2 𝑥= log 9 𝑈𝑠𝑖𝑛𝑔 𝐶ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝐵𝑎𝑠𝑒 𝐹𝑜𝑟𝑚𝑢𝑙𝑎 𝐱 ≈ 𝟎. 𝟑𝟏𝟓𝟓 Barnett/Ziegler/Byleen Business Calculus 12e 30 Solving Exponential Equations Method 2: • Isolate the exponential part on one side, then take the log or ln of both sides of the equation. • Then evaluate. 9𝑥 = 2 log 9𝑥 = log 2 x ∙ log 9 = log 2 log 2 𝑥= log 9 𝐱 ≈ 𝟎. 𝟑𝟏𝟓𝟓 Barnett/Ziegler/Byleen Business Calculus 12e 31 Solving Exponential Equations Solve and round answer to 4 decimal places: 5𝑒 𝑥 = 2 2 𝑥 𝑒 = 5 2 𝑙𝑛 𝑒 = 𝑙𝑛 5 2 1 𝑥 ∙ 𝑙𝑛 𝑒 = 𝑙𝑛 5 2 𝑥 = 𝑙𝑛 5 𝒙 ≈ −𝟎. 𝟗𝟏𝟔𝟑 𝑥 Barnett/Ziegler/Byleen Business Calculus 12e 32 33 Chapter 2 Functions and Graphs Section 6 Logarithmic Functions (Part III) Learning Objectives for Section 2.6 Logarithmic Functions The student will be able to: • Solve applications involving logarithms. Barnett/Ziegler/Byleen Business Calculus 12e 35 Application: Finance How long will it take money to double if compounded (12∙𝑡) monthly at 4% interest? 0.04 𝑟 𝐴 =𝑃 1+ 𝑛 ln 2 = ln 1 + 𝑛𝑡 0.04 2𝑃 = 𝑃 1 + 12 0.04 2= 1+ 12 (12∙𝑡) (12∙𝑡) You can take the log or the ln of both sides. 12 0.04 ln 2 = 12t ∙ ln 1 + 12 ln 2 =t 0.04 12 ∙ ln 1 + 12 𝑡 ≈ 17.4 It will take about 17.4 yrs for the money to double. Barnett/Ziegler/Byleen Business Calculus 12e 36 Application: Finance Suppose you invest $1500 into an account that is compounded continuously. At the end of 10 years, you want to have a balance of $6500. What must the annual percentage rate be? 13 ln 𝐴 = 𝑃𝑒 𝑟𝑡 6500 = 1500𝑒 (𝑟∙10) 6500 = 𝑒 (𝑟∙10) 1500 13 ln = ln 𝑒 3 𝑟∙10 Barnett/Ziegler/Byleen Business Calculus 12e 3 = 10𝑟 ∙ ln 𝑒 13 ln 3 =𝑟 10 𝑟 ≈ .147 The annual percentage rate must be 14.7% 37 Application: Archeology Recall from Lesson 2-5 that Carbon-14 decays according to the model: 𝐴 = 𝐴0 𝑒 −0.000124𝑡 Estimate that age of a fossil if 15% of the original amount of C-14 is still present. 0.15 = 1 ∙ 𝑒 (−0.000124∙𝑡) ln 0.15 = ln 𝑒 ln 0.15 =𝑡 −0.000124 −0.000124∙𝑡 ln 0.15 = −0.000124𝑡 ∙ ln 𝑒 Barnett/Ziegler/Byleen Business Calculus 12e 𝑡 ≈ 15,299 The fossil would be 15,299 years old. 38 Application: Sound Intensity Sound intensity is measured using the formula: 𝐼 = 𝐼0 ∙ 10𝑁 10 I = sound intensity in watts per 𝑐𝑚2 𝐼0 = intensity of sound just below the threshold of hearing = 10−16 𝑊/𝑐𝑚2 N = number of decibels Barnett/Ziegler/Byleen Business Calculus 12e 39 Application: Sound Intensity Solve for N: 𝐼 = 𝐼0 ∙ 10𝑁 10 𝐼 = 10𝑁 𝐼0 𝐼 𝑙𝑜𝑔 𝐼0 10 = log(10𝑁 𝐼 𝑙𝑜𝑔 = 𝑁 10 𝐼0 𝐼 𝑁 = 10 ∙ 𝑙𝑜𝑔 𝐼0 10 ) 𝐼 𝑙𝑜𝑔 = 𝑁 10 log 10 𝐼0 Barnett/Ziegler/Byleen Business Calculus 12e 40 Application: Sound Intensity Use the formula from the previous example to find the number of decibels for the sound of heavy traffic which has a sound intensity of 10−8 𝑊/𝑐𝑚2 𝐼 𝑁 = 10 ∙ 𝑙𝑜𝑔 (𝐼0 = 10−16 𝑊/𝑐𝑚2 ) 𝐼0 10−8 𝑁 = 10 ∙ 𝑙𝑜𝑔 −16 10 𝑁 = 10 ∙ 𝑙𝑜𝑔108 𝑁 = 10 ∙ 8 ∙ log 10 The sound of heavy traffic is 𝑁 = 80 about 80 decibels. Barnett/Ziegler/Byleen Business Calculus 12e 41 Logarithmic Regression When the scatter plot of a data set indicates a slowly increasing or decreasing function, a logarithmic function often provides a good model. We use logarithmic regression on a graphing calculator to find the function of the form y = a + b*ln(x) that best fits the data. Barnett/Ziegler/Byleen Business Calculus 12e 42 Example of Logarithmic Regression A cordless screwdriver is sold through a national chain of discount stores. A marketing company established the following price-demand table, where x is the number of screwdrivers in demand each month at a price of p dollars per screwdriver. x p = D(x) 1,000 2,000 3,000 4,000 5,000 91 73 64 56 53 Barnett/Ziegler/Byleen Business Calculus 12e Find a log regression equation to predict the price per screwdriver if the demand reaches 6,000. 43 Example of Logarithmic Regression x p = D(x) 1,000 2,000 3,000 4,000 5,000 91 73 64 56 53 𝑦 = 256.47 − 24.04(ln 𝑥) Barnett/Ziegler/Byleen Business Calculus 12e 44 Example of Logarithmic Regression Xmax=6500 Trace Up arrow Enter 6000 𝑊ℎ𝑒𝑛 𝑡ℎ𝑒 𝑑𝑒𝑚𝑎𝑛𝑑 𝑖𝑠 6000, 𝑡ℎ𝑒 𝑝𝑟𝑖𝑐𝑒 𝑤𝑖𝑙𝑙 𝑏𝑒 𝑎𝑏𝑜𝑢𝑡 $47.35 Barnett/Ziegler/Byleen Business Calculus 12e 45 46