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Chapter 8

Chapter 8. Exponential and Logarithmic Functions. In this chapter, you will …. Learn to use exponential functions to model real-world data. Learn to graph exponential functions and their inverses, logarithmic functions. Learn to solve exponential and logarithmic equations.

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Chapter 8

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  1. Chapter 8 Exponential and Logarithmic Functions

  2. In this chapter, you will … • Learn to use exponential functions to model real-world data. • Learn to graph exponential functions and their inverses, logarithmic functions. • Learn to solve exponential and logarithmic equations

  3. 8-1 Exploring Exponential Models • What you’ll learn … • To model exponential growth • To model exponential decay 2.03 Use exponential functions to model and solve problems; justify results. • Solve using tables, graphs, and algebraic properties. • Interpret the constants and coefficients in the context of the problem. 2.04 Create and use best-fit mathematical models of linear, exponential, and quadratic functions to solve problems involving sets of data. • Interpret the constants, coefficients, and bases in the context of the data. • Check the model for goodness-of-fit and use the model, where appropriate, to draw conclusions or make predictions.

  4. An exponential function is a function with the general form of y = abx where x is a real number, a ≠ 0, b > 0, and b ≠ 1. You can use an exponential function to model growth when b > -1. When b > 1, b is the growth factor.

  5. Example 1 Graphing Exponential Growth y = 2x

  6. Real World Connection Refer to the graph. In 2000, the annual rate of increase in the US population was about 1.24%. • Find the growth factor for the US population. • Suppose the rate of increase continues to be 1.24%. Write a function to model US population growth. Population (millions) Year

  7. Write an exponential function y = abx for a graph that includes (2,2) and (3,4). Example 3 Writing an Exponential Function Write an exponential function y = abx for a graph that includes (2,4) and (3,16).

  8. An exponential function can be used to model decay, when 0 < b < 1. When b < 1, b is the decay factor.

  9. Example 4 Analyzing a Function Without graphing, determine whether the function y = 14(0.95)x represents exponential growth or exponential decay. Without graphing, determine whether the function y = 0.2(5)x represents exponential growth or exponential decay.

  10. An asymptote is a line that a graph approaches as x or y increases in absolute value.

  11. Example 5a Graphing Exponential Decay y = 24(1/3)x Identify. • Critical point • Horizontal asymptote • Domain • range

  12. Example 5b Graphing Exponential Decay y = 100(0.1)x Identify. • Critical point • Horizontal asymptote • Domain • range

  13. Example 6 Real World Connection The exponential decay graph shows the expected depreciated for a car over four years. Estimate the value of the car after 6 years. The decay factor b = 1 + r, where r is the annual rate of decrease. The initial value of the car is $20,000. After one year the value of the car is about $17,000. 20,000 15,000 10,000 5,000 Value ($) 0 1 2 3 4 Years since purchase

  14. 8-2 Properties of Exponential Functions • What you’ll learn … • To identify the role of constants in y=abcx • To use e as a base 2.03 Use exponential functions to model and solve problems; justify results. • Solve using tables, graphs, and algebraic properties. • Interpret the constants and coefficients in the context of the problem.

  15. So far we have graphed functions of the form y = abx for values of a greater than 0. When a < 0, the graph of y = abx is a reflection over the x-axis.

  16. Graph and give asymptote, critical point, domain and range. y = ½ 2x y = (1/3)x *

  17. You can graph many exponential functions as translations of the parent function y = abx. • The graph of y = abx+k + d is the graph of y = abx translated k units left or right and d units up or down.

  18. Example 2 Translating y = abx y =8(1/2)x y = 8(1/2)x+2 +3

  19. Example 2b Translating y = abx y =2(3)x-1 + 1 y = -3(4)x+1 +2

  20. Half-life What does that mean? • The half-life is the amount of time it takes for half of the atoms in a sample to decay. A = A0(1/2) t/k where k is half-life.

  21. Example 3 Real World Connection A hospital prepares a 100-mg supply of technetium-99m, which has a half-life of 6 hours. Make a table showing the amount of technetium-99m that remains at the end of each 6-hour interval for 36 hours. Then write an exponential function to find the amount of technetium-99m that remains after 75 hours. Relate The amount of technetium-99m is an exponential function of the number of half-lives. The initial amount is 100 mg. The decay factor is ½. One half-life equals 6 hours. Define Let y = the amount of technetium-99m. Let x = the number of hours elapsed. Then 1/6x = the number of half-lives.

  22. The Number …. e? • Exponential functions with a base of e are useful for describing continuous growth or decay. Your graphing calculator has a key for ex. ≈ 2.271828

  23. Graph y = ex • Find e3 • Find e-3 • Find e1/4 • Find e√x

  24. In previous courses you have studied simple interest and compound interest. Compound Interest Simple Interest

  25. The more frequently interest is compounded, the more quickly the amount in an account increases. The formula for continuously compounded interest uses the number e.

  26. Suppose you invest $1050 at an annual interest rate of 5.5% compounded continuously. How much will you have in the account after 5 years? Example 5 Real World Connections A = Pert Suppose you invest $1300 at an annual interest rate of 4.3% compounded continuously. How much will you have in the account after three years?

  27. 8-3 Logarithmic Functions as Inverses • What you’ll learn … • To write and evaluate logarithmic expressions • To graph logarithmic functions 1.01 Simplify and perform operations with rational exponents and logarithms (common and natural) to solve problems. 2.01 Use the composition and inverse of functions to model and solve problems; justify results.

  28. The exponents used by the Richter scale are called logarithms or logs. Definition Logarithm The logarithm to the base b of a positive number y is defined as follows: If y = bx, then logb y = x.

  29. Example 2 Writing in Logarithmic Form • Write 25 = 52 • Write 729 = 36 • Write (1/2)3 = 1/8 • Write 100 = 1

  30. Example 3 Evaluating Logarithms • Evaluate log8 16 • Evaluate log9 27 • Evaluate log64 1/32 • Evaluate log10 100

  31. A common logarithm is a logarithm that uses base 10. You can write the common logarithm log10 y as log y. • Scientist use common logarithms to measure acidity, which increases as the concentration of hydrogen ions in a substance increases. The pH of a substance equals –log[H+], where [H+] is the concentration of hydrogen ions.

  32. Example 4 Real World Connection The pH of lemon juice is 2.3, while the pH of milk is 6.6. Find the concentration of hydrogen ions in each substance. Which substance is more acidic?

  33. A logarithmic function is the inverse of an exponential function.

  34. Example 5a Graphing a Logarithmic Function Graph y = log2 x. Step 1 Graph y=2x. Step 2 Draw y=x. Step 3 Choose points y=2x. Then reverse the coordinates and plot the points of y = log2 x.

  35. Example 5b Graphing a Logarithmic Function Graph y = log3 x. Step 1 Graph y=3x. Step 2 Draw y=x. Step 3 Choose points y=3x. Then reverse the coordinates and plot the points of y = log3 x.

  36. Translations of Logarithmic Functions

  37. Graph y=log6(x-2)+3 Graph y=log3 x Example 6 Translating y=logbx

  38. 8-4 Properties of Logarithms What you’ll learn … • To use the properties of logarithms 1.01 Simplify and perform operations with rational exponents and logarithms (common and natural) to solve problems.

  39. Simplify • log2 4 + log2 8 • log3 9 + log3 27 • log2 16 + log2 64

  40. Investigation • Complete the table. Round to the nearest thousandth. • Complete each pair of statements. What do you notice? • log 3+log 5= ___ and log (3 5) = ____ • log 1 + log 7 = ___ and log (1 7) = ____ • log 2 + log 4 = ___ and log (2 4) = ____ • log 10 + log 2 = ____ and log (10 2) = ____ * * * *

  41. Investigation continued • Complete the statement: log M + log N = __________. • A. Make a conjecture. How could you rewrite the expression log using the expressions log M and log N? B. Use your calculator to verify your conjecture for several values of M and N. M N

  42. Properties of Logarithms For any positive numbers M, N, and b, b ≠1, logb MN = logb M + logb N Product Property logb = logb M - logb N Quotient Property logb Mx = x logb M Power Property M N

  43. Example 1 Identifying the Properties of Logarithms • log2 8 – log2 4 = log2 2 • log5 2 + log5 6 = log5 12 • 3 logb 4 – 3 logb 2 = logb 8

  44. Example 2 Simplifying Logarithms • log3 20 – log3 4 • 3 log 2 + log4 – log 16

  45. In Class • Page 449 1-17 odd

  46. Example 3 Expanding Logarithms x y • log5 • log2 7b • log 2 y 3

  47. In Class • Page 449 19 - 29 odd

  48. Logarithms are used to model sound. The intensity of a sound is a measure of the energy carried by the sound wave. The greater the intensity of a sound, the louder it seems. This apparent loudness L is measured in decibels. You can use the formula L = 10 log , where I is the intensity of the sound in watts per square meter. I0 is the lowest intensity sound that the average human ear can detect. I I0

  49. 8-5 Exponential and Logarithmic Equations What you’ll learn … • To solve exponential equations • To solve logarithmic equations 1.01 Simplify and perform operations with rational exponents and logarithms (common and natural) to solve problems. 2.01 Use the composition and inverse of functions to model and solve problems; justify results.

  50. Evaluate * • log9 81 log9 3 • log 10 log3 9 • log2 16 ÷ log2 8 • Simplify 125-2/3 *

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