Honors Algebra 2 Chapter 8 Page 1

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1 Section 8.1 Eponential Functions Goals: 1. To simplify epressions and solve eponential equations involving real eponents. I. Definition of Eponential Function An function is in the form, where and. II. Important Information A. General Equation: B. Initial Value C. Growth vs. Decay 1. Growth a) b). Decay a) b) y = ab 8 6 III. Graphing Eponential Functions A. Graph the growth function: y = 1. What is the domain and range of the function?. Any asymptotes? B. Graph the decay function: y = 5 1. What is the domain and range of the function?. Any asymptotes? IV. Growth or Decay A. Determine whether the following are eponential growth or decay. 1. y = ( 0.7) 1. y = ( ). y = 10. y = ( 0.5) 1 y = ( ) y = 10 5 Honors Algebra Chapter 8 Page 1

2 B. Applications 1. In 006, there were 1,00,000,000 people worldwide using the Internet. At that time, the number of users was growing by 19.5% annually. Write an equation representing the number of users from 006 to 016, if that rate continued.. The pressure of the atmosphere is 1.7 lb/in at Earth s surface. It decreases by about 0% for each mile of altitude up to about 50 miles. Write an equation that estimates the atmospheric pressure at an altitude of t miles. Estimate the atmospheric pressure at an altitude of 10 miles.. The pressure of a car tire with a bent rim is.7 lb/in at the start of a road trip. It decreases by about % for each mile driven due to a leaky seal. Write an equation that estimates the air pressure of the tire after t miles. Estimate the air pressure of the tire after 0 miles. Homework: Worksheet Honors Algebra Chapter 8 Page

3 Honors Algebra Worksheet Section 8.1 Honors Algebra Chapter 8 Page

4 Honors Algebra Chapter 8 Page

5 Section 8.1B Modeling using Eponential Functions V. Work Together A. In previous lessons you have learned how to find a line of best fit for a set of data. Some data sets are better modeled by eponential functions. 1. Draw a scatter plot for the given data [0,18] scale: 1 [0,800] scale: 50 \ US Sales of Compact Disks Year Millions of CD's Find the eponential regression equation (EpReg) that best fits the data and graph.. Write a sentence that describes the equation and data. 5. Based on the graph find the number of CDs sold in 000. B. Wrap Up 6. Do you think a linear regression line would be a good model for the data? Why or why not? (Hint: Eplain the difference between the two regression curves.) Homework: Worksheet Honors Algebra Chapter 8 Page 5

6 Years Since 1780 Worksheet Section 8.1B: Modeling using Eponential Functions Name: Population (In millions) Years Since 1780 Population (In millions) Years Since 1780 Population (In millions) Use a graphing calculator to draw a scatter plot of the data. Then calculate and graph the curve of best fit that shows how the year is related to the population. Use EpReg for this problem. Use: [0.00] scale:0 [0,00] scale:0. Write the equation of best fit.. Write a sentence that describes the equation and data Based on the graph, estimate the population for 000. Eplain how you found your answer Based on the graph, when will the population reach 5 million? Eplain how you found your answer. Honors Algebra Chapter 8 Page 6

7 Section 8. Solving Eponential Equations and Inequalities Objectives: 1. To solve eponential equations and inequalities. 1 I. Solving Eponential Equations in the form a = a A. Property: Let a > 0 and a 1. Then iff. B. Eamples: = 9. =. = 9 5 C. Writing an eponential function: Eample: 1. In 000, the population of Phoeni was 1,1,05. By 007, it was estimated at 1,51,986. Write an eponential function that could be used to model the population of Phoeni. Write t in terms of the numbers of years since In 000, the population of Phoeni was 1,1,05. By 007, it was estimated at 1,51,986. Predict the population of Phoeni in 01. II. Solving Eponential Inequalities in the form A. Property: Let a > 1. Then iff and iff. B. Eamples: 1. 5 > > 1 Homework: p all, 16, 17,, (5-7)/ Honors Algebra Chapter 8 Page 7

8 Section 8. Logarithms and Logarithmic Functions Objectives: 1. Evaluate logarithmic functions.. Graph logarithmic functions I. Logarithmic Functions and Epressions A. Write the following in Inverse form: 1. y =. y =. y = +. y = B. Important Information 1. Inverse function of eponential functions = b. Notation: If, then.. Equivalence Statement:. Question asked for by the symbol log b : b to what will equal? C. Eamples: 1. Logarithmic to Eponential Form a) log 9 = b) log = c) log 8 = 1 y y = y = log. Eponential to Logarithmic Form a) 5 = 15 b) 1 7 = c) = 81. Evaluate Logarithmic Epressions a) log b) log II. Graphing Logarithmic Functions A. Eamples: f = log 1. ( ) Honors Algebra Chapter 8 Page 8

9 f = log. f ( ) = log 1. ( ) B. Transformations of Logarithmic Functions Eamples: 1 log 1 1. f ( ) = 6. f ( ) = log ( + ) Homework: p all, (15-5)/, 9, 57, 65, 7, 7 Honors Algebra Chapter 8 Page 9

10 Section 8. Solving Logarithmic Equations and Inequalities Objectives: 1. To solve equations involving logarithms.. To solve inequalities involving logarithms. I. Logarithmic Equations Eamples: Solve the following Logarithmic Equations 1. log8 =. log7 n =. log = log ( 6 8). log = log ( + 0) II. Logarithmic Inequalities A. Property 1 B. Eamples 1. log6 >. log < C. Property D. Eamples: log 8 log 5 1. ( + ) > ( + ). log ( + 5) < log ( 5 + 1) Homework: p all, 15, 16, 5,,, 8, (51-66)/ Honors Algebra Chapter 8 Page 10

11 Section 8.5 Properties of Logarithms Objectives: 1. To simplify and evaluate epressions using properties of logarithms. To solve equations involving logarithms I. Logarithmic Properties A. Properties: p 1. log b mn =. log m = p log m m. log b n =. log log b m = b or b m = b B. Eamples: Given log = and log 5 = Find: log 8. Find: log 9 b b. Find: log 5. Find: log 6. Find: log 6 6. Find: log 15 C. Solve 1. log ( + 5) log ( ) =. log ( + ) log ( 1) = log log8 = log8 81 Homework: p all, 6-11 all, 1-17 all, 19-1 all, all Honors Algebra Chapter 8 Page 11

12 Section 8.6 Common Logarithms Objectives: 1. To find common logarithms and antilogarithms.. To solve problems involving common logarithms.. To solve eponential equations and inequalities. I. The common logarithm(log) is log 10 Eamples 1. log 1000 =. log 100 =. log 0.01 =. log 7 = II. Antilogarithms A. Antilogarithm means to apply the inverse of a logarithm. B. What is the inverse of y = log10? C. Eamples 1. = log = log. The loudness L, in decibels, of a sound is L 10log I = where I is the intensity of m the sound and m is the minimum intensity of sound detectable by the human ear. The sound of a jet engine can reach a loudness of 15 decibels. How many times the minimum intensity of audible sound is this, if m is defined to be 1? III. Solve Eponential Equations and Inequalities Using Logarithms Eamples 1. 5 = 6. = >. 5 < 10 IV. log m a Converting logarithms: logb a = log m b Eamples: 1. log510. log5 16 Homework: p all, 9, (-51)/, 61, 68, odds Honors Algebra Chapter 8 Page 1

13 Section 8.7 Base e and Natural Logarithms Objectives; 1. To evaluate epressions involving the natural base and natural logarithm.. To solve eponential equations and inequalities using natural logarithms.. To solve problems involving natural logarithms and e. I. Natural Base Functions A. Key Concept B. Equivalent Statement: C. Eamples: 1. e =. e =. ln ln 5 = II. Natural Logarithmic Properties A. Same as any other logarithm B. Eamples: Write as a single natural logarithm 1. ln + ln 6. ln + ln + ln y III. Solve Eponential Equations and Inequalities Using Natural Logarithms A. Eamples e + =. ln 5 6 =. ( ) ln + 1 > 8 Honors Algebra Chapter 8 Page 1

14 B. Investments Compounded Continuously Eamples 1. Suppose you deposit $700 into an account paying an APR of %, compounded continuously. What is the balance after 8 years?. Suppose you deposit $700 into an account paying an APR of %, compounded continuously. How long will it take for the balance in your account to reach at least $100?. Suppose you deposit an unknown amount into an account paying an APR of %, compounded continuously. How much would have to be deposited in order to reach a balance of $1500 after 1 years? Homework: p all, 0, 7, 56, (68-81)/ Honors Algebra Chapter 8 Page 1

15 Section 8.8 Using Eponential and Logarithmic Functions Objectives: 1. To use logarithms to solve problems involving eponential growth and decay.. To use logarithms to solve problems involving logistic growth. I. Continuous Eponential Growth and Decay A. The Function B. Eamples 1. The half-life of Sodium- is.6 years. a) Determine the rate of decay for Sodium-. b) A geologist eamining a meteorite estimates that it contains only about 10% as much Sodium- as it would have contained when it reached the surface of the Earth. How long ago did the meteorite reach the surface of the Earth?. In 007, the population of China was 1. billion. In 000, it was 1.6 billion. a) Determine China s relative rate of growth. b) When will China s population reach 1.5 billion? c) India s population in 007 was 1.1 billion and can be modeled by 0.015t y = 1.1e. Determine when India s population will surpass China s. Honors Algebra Chapter 8 Page 15

16 II. Logistic Growth Function A. The Function B. Eamples A city s population in millions is modeled by ( ) 0. of years since Graph the function. f t 1. =, where t is the number t e What is the horizontal asymptote?. What will be the maimum population?. According to the function, when will the city s population reach 1 million? Homework: p all, 10, 1, 5, 6 Honors Algebra Chapter 8 Page 16