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Chapter 3 Exponential, Logistic, and Logarithmic Functions Quick Review Evaluate the expression without using a calculator. 1. -125 3 27 64 3. 27 Rewrite the expression using a single positive exponent. 2. 3 4/3 4. a -3 2 Use a calculator to evaluate the expression. 5. 3.71293 5 Slide 3- 2 Quick Review Solutions Evaluate the expression without using a calculator. 1. -125 3 -5 27 3 64 4 3. 27 81 Rewrite the expression using a single positive exponent. 2. 3 4/3 1 a Use a calculator to evaluate the expression. 4. a -3 2 5. 3.71293 5 Slide 3- 3 6 1.3 Exponential Functions Let a and b be real number constants. An exponential function in x is a function that can be written in the form f ( x) a b , where a is nonzero, x b is positive, and b 1. The constant a is the initial value of f (the value at x 0), and b is the base. Slide 3- 4 Determine if they are exponential functions • • • • • ℎ = 4 = 6 −9 = −2 ∗ 1.5 = 7 ∗ 3− = 5 ∗ 63 Answers • • • • • Yes No Yes Yes no Sketch an exponential function Example Finding an Exponential Function from its Table of Values Determine formulas for the exponential function g and h whose values are given in the table below. Slide 3- 8 Example Finding an Exponential Function from its Table of Values Determine formulas for the exponential function g and h whose values are given in the table below. Because g is exponential, g ( x) a b . Because g (0) 4, a 4. x Because g (1) 4 b 12, the base b 3. So, g ( x) 4 3 . 1 x Because h is exponential, h( x) a b . Because h(0) 8, a 8. x 1 Because h(1) 8 b 2, the base b 1/ 4. So, h( x) 8 . Slide 3- 9 4 x 1 Exponential Growth and Decay For any exponential function f ( x) a b and any real number x, f ( x 1) b f ( x). x If a 0 and b 1, the function f is increasing and is an exponential growth function. The base b is its growth factor. If a 0 and b 1, the function f is decreasing and is an exponential decay function. The base b is its decay factor. Slide 3- 10 Sketch exponential graph and determine if they are growth or decay • = 2 • = 1 3 • ℎ = 4− Example Transforming Exponential Functions Describe how to transform the graph of f ( x) 2 into the graph of g ( x) 2 . x x -2 Slide 3- 12 Example Transforming Exponential Functions Describe how to transform the graph of f ( x) 2 into the graph of g ( x) 2 . x x -2 The graph of g ( x) 2 is obtained by translating the graph of f ( x) 2 by x -2 x 2 units to the right. Slide 3- 13 Example Transforming Exponential Functions Describe how to transform the graph of f ( x) 2 into the graph of g ( x) 2 . x -x The graph of g ( x) 2 is obtained by reflecting the graph of f ( x) 2 across x x the y-axis. Slide 3- 14 Group Activity • Use this formula 1 + • • • • • • • • 1 Group 1 calculate when x=1 Group 2 calculate when x=2 Group 3 calculate when x=4 Group 4 calculate when x=12 Group 5 calculate when x=365 Group 6 calculate when x=8760 Group 7 calculate when x=525600 Group 8 calculate when x=31536000 • What do you guys notice? The Natural Base e 1 e lim 1 x x Slide 3- 16 x Exponential Functions and the Base e Any exponential function f ( x) a b can be rewritten as f ( x) a e , x for any appropriately chosen real number constant k . If a 0 and k 0, f ( x) a e is an exponential growth function. If a 0 and k 0, f ( x) a e is an exponential decay function. kx kx Slide 3- 17 kx Exponential Functions and the Base e Slide 3- 18 Example Transforming Exponential Functions Describe how to transform the graph of f ( x) e into the graph of g ( x) e . x 3x Slide 3- 19 Example Transforming Exponential Functions Describe how to transform the graph of f ( x) e into the graph of g ( x) e . x 3x The graph of g ( x) e is obtained by horizontally shrinking the graph of 3x f ( x) e by a factor of 3. x Slide 3- 20 Logistic Growth Functions Let a, b, c, and k be positive constants, with b 1. A logistic growth function in x is a function that can be written in the form f ( x) f ( x) Slide 3- 21 c 1 a e kx c or 1 a b where the constant c is the limit to growth. x Example: Graph and Determine the horizontal asymptotes • = 7 1+3∗.6 Answer • Horizontal asymptotes at y=0 and y=7 • Y-intercept at (0,7/4) Group Work: Graph and determine the horizontal asymptotes • () = 26 1+2 −4 Answer • Horizontal asymptotes y=0 and y=26 • Y-intercept at (0,26/3) Word Problems: • Year 2000 • Year 2010 782,248 people 923,135 people • Use this information to determine when the population will surpass 1 million people? (hint use exponential function) Group Work • Year 1990 • Year 2000 156,530 people 531,365 people • Use this information and determine when the population will surpass 1.5 million people? Word Problem • The population of New York State can be modeled by • = 19.875 1+57.993 −0.035005 • ℎ ℎ 1800 • A) What’s the population in 1850? • B) What’s the population in 2010? • C) What’s the maximum sustainable population? Answer • A) 1,794,558 • B) 19,161,673 • C) 19,875,000 Group Work In chemistry, you are given half-life formulas = 0 = ℎ , = , 0 = If you are given a certain chemical have a halflife of 56.3 minutes. If you are given 80 g first, when will it become 16 g? Homework Practice • P 286 #1-54 eoe EXPONENTIAL AND LOGISTIC MODELING Review • We learned that how to write exponential functions when given just data. • Now what if you are given other type of data? That would mean some manipulation Quick Review Convert the percent to decimal form or the decimal into a percent. 1. 16% 2. 0.05 3. Show how to increase 25 by 8% using a single multiplication. Solve the equation algebraically. 4. 20 b 720 Solve the equation numerically. 5. 123 b 7.872 2 3 Slide 3- 34 Quick Review Solutions Convert the percent to decimal form or the decimal into a percent. 1. 16% 0.16 2. 0.05 5% 3. Show how to increase 25 by 8% using a single multiplication. Solve the equation algebraically. 25 1.08 4. 20 b 720 6 Solve the equation numerically. 5. 123 b 7.872 0.4 2 3 Slide 3- 35 Exponential Population Model If a population P is changing at a constant percentage rate r each year, then P(t ) P (1 r ) , where P is the initial population, r is expressed as a decimal, t 0 and t is time in years. Slide 3- 36 0 Example: • You are given = 300 1.05 • Is this a growth or decay? What is the rate? Example Finding Growth and Decay Rates Tell whether the population model P(t ) 786,543 1.021 is an exponential t growth function or exponential decay function, and find the constant percent rate of growth. Slide 3- 38 Example • You are given = 800 .15 • Is this a growth or decay? What is the rate? Example Finding an Exponential Function Determine the exponential function with initial value=10, increasing at a rate of 5% per year. Slide 3- 40 Group Work • Suppose 50 bacteria is put into a petri dish and it doubles every hour. When will the bacteria be 350,000? Answer • = 50 2 • 350000 = 50 2 • t=12.77 hours Example Modeling Bacteria Growth Suppose a culture of 200 bacteria is put into a petri dish and the culture doubles every hour. Predict when the number of bacteria will be 350,000. Slide 3- 43 Group Work: half-life • Suppose the half-life of a certain radioactive substance is 20 days and there are 10g initially. Find the time when there will be 1 g of the substance. answer • Just the setting up • = • 1 = 10 1 2 ℎ 20 Group Work • You are given = 150 1.025 • When will this become 150000? Example Modeling U.S. Population Using Exponential Regression Use the 1900-2000 data and exponential regression to predict the U.S. population for 2003. Slide 3- 47 Example Modeling a Rumor A high school has 1500 students. 5 students start a rumor, which spreads logistically so that S (t ) 1500 /(1 29 e ) models the number of students who have heard the rumor by the end of t days, where t 0 is the day the -0.9 t rumor begins to spread. (a) How many students have heard the rumor by the end of Day 0? (b) How long does it take for 1000 students to hear the rumor? Slide 3- 48 Example Modeling a Rumor: Answer A high school has 1500 students. 5 students start a rumor, which spreads logistically so that S (t ) 1500 /(1 29 e ) models the number of students who have heard the rumor by the end of t days, where t 0 is the day the -0.9 t rumor begins to spread. (a) How many students have heard the rumor by the end of Day 0? (b) How long does it take for 1000 students to hear the rumor? (a) S (0) 1500 /(1 29 e -0.9 ( 0 ) ) 1500 /(1 29 1) 1500 / 30 50. So 50 students have heard the rumor by the end of day 0. (b) Solve 1000 1500 /(1 29 e ) for t. -0.9 t t 4.5. So 1000 students have heard the rumor half way through the fifth day. Slide 3- 49 Key Word • Maximum sustainable population • What does this mean? What function deals with this? Maximum Sustainable Population Exponential growth is unrestricted, but population growth often is not. For many populations, the growth begins exponentially, but eventually slows and approaches a limit to growth called the maximum sustainable population. Slide 3- 51 Homework Practice (Do in class also) • P 296 #1-44 eoo LOGARITHMIC FUNCTION, GRAPHS AND PROPERTIES Quick Review Evaluate the expression without using a calculator. 1. 6 8 2. 2 3. 7 Rewrite as a base raised to a rational number exponent. 1 4. e -2 11 32 0 3 5. 10 4 Slide 3- 54 Quick Review Solutions Evaluate the expression without using a calculator. 1 1. 6 36 8 2. 2 2 3. 7 1 -2 11 32 0 Rewrite as a base raised to a rational number exponent. 1 4. e 3 5. 10 4 e 3 / 2 1/ 4 10 Slide 3- 55 Changing Between Logarithmic and Exponential Form If x 0 and 0 b 1, then y log ( x) if and only if b x. y b Slide 3- 56 Group Work: transform logarithmic form into exponential form • A)5 25 = 2 • B)3 3 = 1 2 • C) 2 60 = • D) = 8 Group Work: convert exponential form into logarithmic form • 5 = 34 • 5 −2 = 1 25 • 40 = 1 • 161 = Inverses of Exponential Functions Slide 3- 59 Basic Properties of Logarithms For 0 b 1, x 0, and any real number y. log 1 0 because b 1. 0 b log b 1 because b b. 1 b log b y y b b logb x Slide 3- 60 x because b b . y y because log x log x. b b An Exponential Function and Its Inverse Slide 3- 61 Common Logarithm – Base 10 • Logarithms with base 10 are called common logarithms. • The common logarithm log10x = log x. • The common logarithm is the inverse of the exponential function y = 10x. Slide 3- 62 Basic Properties of Common Logarithms Let x and y be real numbers with x 0. log1 0 because 10 1. log10 1 because 10 10. 0 1 log10 y because 10 10 . 10 x because log x log x. y log x Slide 3- 63 y y Example Solving Simple Logarithmic Equations Solve the equation by changing it to exponential form. log x 4 Slide 3- 64 Example Solving Simple Logarithmic Equations Solve the equation by changing it to exponential form. log x 4 x 10 10, 000 4 Slide 3- 65 Basic Properties of Natural Logarithms Let x and y be real numbers with x 0. ln1 0 because e 1. 0 ln e 1 because e e. ln e y because e e . e x because ln x ln x. 1 y y y ln x Slide 3- 66 Graphs of the Common and Natural Logarithm Slide 3- 67 Example Transforming Logarithmic Graphs Describe how to transform the graph of y ln x into the graph of h( x) ln(2 x). Slide 3- 68 Example Transforming Logarithmic Graphs Describe how to transform the graph of y ln x into the graph of h( x) ln(2 x). h( x) ln(2 x) ln[( x 2)]. So obtain the graph of h( x) ln(2 - x) from y ln x by applying, in order, a reflection across the y -axis followed by a translation 2 units to the right. Slide 3- 69 Quick Review Evaluate the expression without using a calculator. 1. log10 2. ln e 3 3 3. log 10 -2 Simplify the expression. 3 3 2 2 xy 4. x y x y 2 5. 4 2x Slide 3- 70 3 1/ 2 Quick Review Solutions Evaluate the expression without using a calculator. 1. log10 2. ln e 3 3 3 3 3. log 10 -2 -2 Simplify the expression. 3 3 2 2 xy 4. x y x y 2 5. x y 4 2x 3 5 5 1/ 2 4 x y 2 2 Slide 3- 71 What you’ll learn about • • • • Properties of Logarithms Change of Base Graphs of Logarithmic Functions with Base b Re-expressing Data … and why The applications of logarithms are based on their many special properties, so learn them well. Slide 3- 72 Properties of Logarithms Let b, R, and S be positve real numbers with b 1, and c any real number. Product rule: log ( RS ) log R log S b Quotient rule: Power rule: Slide 3- 73 b b R log log R log S S log ( R) c log R b b c b b b Example Proving the Product Rule for Logarithms Prove log ( RS ) log R log S. b Slide 3- 74 b b Example Proving the Product Rule for Logarithms Prove log ( RS ) log R log S. b b b Let x log R and y log S . The corresponding exponential statements b b are b R and b S . Therefore, x y RS b b RS b x y x y log ( RS ) x y b change to logarithmic form log ( RS ) log R log S b Slide 3- 75 b b Example Expanding the Logarithm of a Product Assuming x is positive, use properties of logarithms to write log 3 x 5 Slide 3- 76 as a sum of logarithms or multiple logarithms. Example Expanding the Logarithm of a Product Assuming x is positive, use properties of logarithms to write log 3 x 5 as a sum of logarithms or multiple logarithms. log 3x log 3 log x 5 5 log 3 5log x Slide 3- 77 Example Condensing a Logarithmic Expression Assuming x is positive, write 3ln x ln 2 as a single logarithm. Slide 3- 78 Example Condensing a Logarithmic Expression Assuming x is positive, write 3ln x ln 2 as a single logarithm. 3ln x ln 2 ln x ln 2 3 x ln 2 3 Slide 3- 79 Group Work • log(7 2 5 ) Group Work • Expand • 17 2 Group Work • Express as a single logarithm • − + 5 Group Work • Express as a single logarithm • 42 + 2 2 5 − 32 Change-of-Base Formula for Logarithms For positive real numbers a, b, and x with a 1 and b 1, log x log x . log b a b a Slide 3- 84 Example Evaluating Logarithms by Changing the Base Evaluate log 10. 3 Slide 3- 85 Example Evaluating Logarithms by Changing the Base Evaluate log 10. 3 log 10 3 Slide 3- 86 log10 1 2.096 log 3 log 3 Solving • 4 = 51 Solving • ln Solving • log 1 Solving • log 5 = log 4 + log( − 3) Solving • 5 56 = Solving • 25+3 = 16 Homework Practice • Pg 317 #1-50 eoe EQUATION SOLVING AND MODELING Quick Review Prove that each function in the given pair is the inverse of the other. 1. f ( x) e and g ( x) ln x 3x 1/ 3 2. f ( x) log x and g ( x) 10 2 x/2 Write the number in scientific notation. 3. 123,400,000 Write the number in decimal form. 4. 5.67 10 5. 8.91 10 Slide 3- 95 8 -4 Quick Review Solutions Prove that each function in the given pair is the inverse of the other. 1. f ( x) e and g ( x) ln x 3x 1/ 3 2. f ( x) log x and g ( x) 10 2 x/2 f ( g ( x)) e 3 ln x1 / 3 e f ( g ( x)) log 10 x/2 ln x 2 x log10 x x Write the number in scientific notation. 3. 123,400,000 1.234 10 8 Write the number in decimal form. 4. 5.67 10 567, 000, 000 8 5. 8.91 10 -4 0.000891 Slide 3- 96 One-to-One Properties For any exponential function f ( x) b , If b b , then u v. For any logarithmic function f ( x) log x, x u v b If log u log v, then u v. b Slide 3- 97 b Example Solving an Exponential Equation Algebraically Solve 40 1/ 2 5. x/2 Slide 3- 98 Example Solving an Exponential Equation Algebraically Solve 40 1/ 2 5. x/2 40 1/ 2 5 x/2 1 1/ 2 8 1 1 2 2 x/2 3 x6 x/2 x/2 Slide 3- 99 divide by 40 3 1 1 8 2 one-to-one property 3 Example Solving a Logarithmic Equation Solve log x 3. 3 Slide 3- 100 Example Solving a Logarithmic Equation Solve log x 3. 3 log x 3 3 log x log10 x 10 3 3 3 x 10 Slide 3- 101 3 Group Work • ln 3 − 2 + ln − 1 = 2 Group Work: Solve for x • 15 1 2 3 =5 Group Work: Solve • − − 2 =5 Group Work: Solve • 1.05 = 8 Orders of Magnitude The common logarithm of a positive quantity is its order of magnitude. Orders of magnitude can be used to compare any like quantities: • A kilometer is 3 orders of magnitude longer than a meter. • A dollar is 2 orders of magnitude greater than a penny. • New York City with 8 million people is 6 orders of magnitude bigger than Earmuff Junction with a population of 8. Slide 3- 106 Note: • In regular cases, how you determine the magnitude is by how many decimal places they differ • In term of Richter scale and pH level, since the number is the power or the exponent, you just take the difference of them. Example: • What’s the difference of the magnitude between kilometer and meter? • It is 3 orders of magnitude longer than a meter Example: • The order of magnitude between an earthquake rated 7 and Richter scale rated 5.5. • The difference of magnitude is 1.5 Group Work • Find the order of magnitude: • Between A dollar and a penny • A horse weighing 500 kg and a horse weighing 50g • 8 million people vs population of 8 Answer • 2 orders of magnitude • 4 orders of magnitude • 6 orders of magnitude Group Work • Find the difference of the magnitude: • Sour vinegar a pH of 2.4 and baking soda pH of 8.4 • Earthquake in India 7.9 and Athens 5.9 Answer • 6 orders of magnitude • 2 orders of magnitude Richter Scale The Richter scale magnitude R of an earthquake is a R log B, where a is the amplitude in micrometers ( m) T of the vertical ground motion at the receiving station, T is the period of the associated seismic wave in seconds, and B accounts for the weakening of the seismic wave with increasing distance from the epicenter of the earthquake. Slide 3- 114 Example: • How many times more severe was the 2001 earthquake in Gujarat, India (1 =7.9) than the 1999 earthquake in Athens, Greece (2 =5.9) Group Work: Show work • How many times more severs was the earthquake in SF (1 = 6.5) than the earthquake in PS (2 = 3.6)? pH In chemistry, the acidity of a water-based solution is measured by the concentration of hydrogen ions in the solution (in moles per liter). The hydrogen-ion concentration is written [H+]. The measure of acidity used is pH, the opposite of the common log of the hydrogen-ion concentration: pH=-log [H+] More acidic solutions have higher hydrogen-ion concentrations and lower pH values. Slide 3- 117 Example: • Sour vinegar has pH of 2.4 and a box of Leg and Sickle baking soda has a pH of 8.4. • A) what are their hydrogen-ion concentration? • B) How many more times greater is the hydrogen-ion concentration of the vinegar than of the baking soda? Group Work • A substance with pH of 3.4 and another with pH of 8.1 • A) what are their hydrogen-ion concentration? • B) How many more times greater is the hydrogen-ion concentration? Newton’s Law of Cooling An object that has been heated will cool to the temperature of the medium in which it is placed. The temperature T of the object at time t can be modeled by T (t ) T (T T )e for an appropriate value of k , where kt m 0 m T the temperature of the surrounding medium, m T the temperature of the object. 0 This model assumes that the surrounding medium maintains a constant temperature. Slide 3- 120 Example Newton’s Law of Cooling A hard-boiled egg at temperature 100ºC is placed in 15ºC water to cool. Five minutes later the temperature of the egg is 55ºC. When will the egg be 25ºC? Slide 3- 121 Example Newton’s Law of Cooling A hard-boiled egg at temperature 100ºC is placed in 15ºC water to cool. Five minutes later the temperature of the egg is 55ºC. When will the egg be 25ºC? Given T 100, T 15, and T (5) 55. 0 m T (t ) T (T T )e m 0 55 15 85e 40 85e 5 k 5 k 40 e 85 40 ln 5k 85 k 0.1507... 5 k Slide 3- 122 kt m Now find t when T (t ) 25. 25 15 85e 10 85e 0.1507 t 0.1507 t 10 ln 0.1507t 85 t 14.2 min . Group Work • A substance is at temperature 96℃ is placed in 16℃. Four minutes later the temperature of the egg is 45℃. Use Newton’s Law of Cooling to determine when the egg will be 20℃ Regression Models Related by Logarithmic Re-Expression • • • • Linear regression: Natural logarithmic regression: Exponential regression: Power regression: Slide 3- 124 y = ax + b y = a + blnx y = a·bx y = a·xb Three Types of Logarithmic ReExpression Slide 3- 125 Three Types of Logarithmic ReExpression (cont’d) Slide 3- 126 Three Types of Logarithmic ReExpression (cont’d) Slide 3- 127 Homework Practice • Pg 331 #1-51 eoe MATHEMATICS OF FINANCE Interest Compounded Annually If a principal P is invested at a fixed annual interest rate r , calculated at the end of each year, then the value of the investment after n years is A P(1 r ) , where r is expressed as a decimal. n Slide 3- 130 Interest Compounded k Times per Year Suppose a principal P is invested at an annual rate r compounded k times a year for t years. Then r / k is the interest rate per compounding period, and kt is the number of compounding periods. The amount A r in the account after t years is A P 1 . k kt Slide 3- 131 Example Compounding Monthly Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years. Slide 3- 132 Example Compounding Monthly Suppose Paul invests $400 at 8% annual interest compounded monthly. Find the value of the investment after 5 years. Let P 400, r 0.08, k 12, and t 5, r A P 1 k kt 0.08 400 1 12 595.9382... So the value of Paul's investment after 5 years is $595.94. 12 ( 5 ) Slide 3- 133 Group Work • Suppose you have $10000, you invest in a place where they give you 12% interest compounded quarterly. Find the value of your investment after 40 years. Compound Interest – Value of an Investment Suppose a principal P is invested at a fixed annual interest rate r. The value of the investment after t years is r A P 1 when interest compounds k times per year, k A Pe when interest compounds continuously. kt rt Slide 3- 135 Example Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years. Slide 3- 136 Example Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years. P 400, r 0.08, and t 5, A Pe rt 400e 0.08 ( 5 ) 596.7298... So Paul's investment is worth $596.73. Slide 3- 137 Group Work • Suppose you have $10000, you invest in a company where they give you 12% interest compounded continuously. Find the value of your investment after 40 years. Annual Percentage Yield A common basis for comparing investments is the annual percentage yield (APY) – the percentage rate that, compounded annually, would yield the same return as the given interest rate with the given compounding period. Slide 3- 139 Example Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY? Slide 3- 140 Example Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY? Let x the equivalent APY. The value after one year is A 3000(1 x). 0.0465 3000(1 x) 3000 1 4 0.0465 (1 x) 1 4 4 4 0.0465 x 1 1 0.047317... 4 The annual percentage yield is 4.73%. 4 Slide 3- 141 Future Value of an Annuity The future value FV of an annuity consisting of n equal periodic payments of R dollars at an interest rate i per compounding period (payment interval) is 1 i 1 n FV R Slide 3- 142 i . Future Value of an Annuity • At the end of each quarter year, Emily makes a $500 payment into the Lanaghan Mutual Fund. If her investments earn 7.88% annual interest compounded quarterly, what will be the value of Emily’s annuity in 20 years? • Remember i=r/k Group Work • You are currently 18 and you want to retire at age 65. You decide to invest in your future. You are putting in $35 month. If your investment earn 12% annual interest compounded monthly, what will the value of your annuity when you retire? Present Value of an Annuity The present value PV of an annuity consisting of n equal payments of R dollars at an interest rate i per period (payment interval) is PV R 1 1 i i n . Slide 3- 145 Example • Mr. Liu bought a new car for $20000. What are the monthly payment for a 5 year loan with 0 down payment if the annual interest rate (APR) is 2.9%? Homework Practice • Pg 341 #2-56 eoe