CHAPTER 7. Logarithmic Functions

Transcription

1 CHAPTER 7 Logarithmic Functions

2 7.1 CHARACTERISTICS OF LOGARITHMIC FUNCTIONS WITH BASE 10 AND BASE E Chapter 7

3 LOGARITHMS Logarithms are a new operation that we will learn. Similar to exponential functions, logarithms give us a way to solve equations that have variables in the exponent. Ex. 120 = 10 y A logarithmic function is a function of the form: y = alog b x where b > 0, b 1, and a 0, and a and b are real numbers.

4 USING A TABLE OF VALUES, PLOT BOTH GRAPHS ON THE SAME GRID. y 10 x x y x 10 x y What do you notice about the two tables of values/graphs? The x and y-values are interchanged. y y x

5 y x DRAW THE LINE SAME GRID. ON THE y What do you notice? x The graphs are reflections of each other

6 y 10 The equation exponential function. x represents an x 10 The equation represents the INVERSE of the exponential function y The equation of the inverse function can be written as a common logarithmic function: y log10 x This reads log base 10 of x and it means what exponent would we have to raise 10 to in order to get x

7 SIMILARITIES AND DIFFERENCES EXPONENTIAL FUNCTION y 10 x x 10 LOGARITHMIC FUNCTION Or y y log10 x Domain Range x-intercept y-intercept Increasing or decreasing? End Behaviour

8 NOTE! The base of a logarithmic function can be some value other than 10, but 10 is the most common value. In fact, log base 10 can be written as: log x Or y 10 y log x It is not necessary to write the base of 10. You will also see this is the case on your calculator.

9 In general, the inverse of an exponential function: y a( b) can be written as a logarithmic function in the form: y alog b x x

10 WHAT IMPACT DOES THE VALUE OF A HAVE ON THE APPEARANCE OF THE GRAPH? f(x) = log 10 x f(x) = 2log 10 x f(x) = -2log 10 x a = 1 a = 2 y a = -2 x

11 CONCLUSION When a>0, The y-values increase as the x-values increase. This is called an increasing function from Quadrant IV to Quadrant I. When a<0, The y-values decrease as the x-values increase. This is called a decreasing function from Quadrant I to Quadrant IV.

12 COMPLETE THE CHART y 3log x 1 x log x 2 y 2.3log x Value of a Increasing or decreasing? End Behaviour Domain Range x-intercept y-intercept

13 NATURAL LOGARITHMS A logarithm with base e is called the natural logarithm and is written as lnx. The functions y = log e x, y = lnx, x = e y are equivalent.

14 WHAT IS BASE E? An Interesting Fact! Jacob Bernoulli discovered this constant by studying a question about compound interest. An account starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value of the account at year-end will be $2.00. What happens if the interest is computed and credited more frequently during the year?

15 If the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial $1 is multiplied by 1.5 twice, yielding $ = $2.25 at the end of the year. Compounding quarterly yields $ = $ , and compounding monthly yields $1.00 (1+1/12) 12 = $ If there are n compounding intervals, the interest for each interval will be 100%/n and the value at the end of the year will be $1.00 (1 + 1/n) n.

16 Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger n and, thus, smaller compounding intervals. Compounding weekly (n = 52) yields $ , while compounding daily (n = 365) yields $ , just two cents more. The limit as n grows large is the number that came to be known as e; with continuous compounding, the account value will reach $

17 USING A TABLE OF VALUES, PLOT BOTH GRAPHS ON THE SAME GRID. y e x y x x e x y y What do you notice about the two tables of values/graphs? The x and y-values are interchanged. y x

18 DRAW THE LINE SAME GRID. y x ON THE y x What do you notice? The graphs are reflections of each other

19 y e The equation exponential function. x represents an x e y The equation represents the INVERSE of the exponential function The equation of the inverse function can be written as a natural logarithmic function: y ln x This reads the natural log of x and it means what exponent would we have to raise e to in order to get x

20 EXAMPLE 1 Predict the x-intercept, the number of y-intercepts, the end behaviour, the domain, and the range of the following function: y = 15lnx Use the equation of the function to make your predictions. Verify your predictions using your graphing calculator.

21 EXAMPLE 2 Predict the x-intercept, the number of y-intercepts, the end behaviour, the domain, and the range of the following function: y = 4lnx Use the equation of the function to make your predictions. Verify your predictions using your graphing calculator.

22 EXAMPLE 3 Which function matches each graph below? i) y = 5(2) x ii) y = 2(0.1) x iii) y = 6logx iv) y = 2lnx

23 YOUR TURN 1. What best describes the graph? A. A decreasing exponential function B. An increasing exponential function C. A decreasing logarithmic function D. An increasing logarithmic function

24 YOUR TURN 2.What best describes the graph? y x A. A decreasing exponential function B. An increasing exponential function C. A decreasing logarithmic function D. An increasing logarithmic function

25 YOUR TURN 3.What best describes the graph? A. A decreasing exponential function B. An increasing exponential function C. A decreasing logarithmic function D. An increasing logarithmic function

26 YOUR TURN 4. What is the x-intercept of 7log x? A. ( 7,0) y 10 B. C. D. (0,0) (1,0) (10,0)

27 YOUR TURN 5. Explain why we cannot evaluate log 10 ( 3) or y ln(0). [Hint! The graphs of the common log and the natural logarithm will help you explain this answer].

28 YOUR TURN 5. Explain why we cannot evaluate log 10 ( 3) or y ln(0). [Hint! The graphs of the common log and the natural logarithm will help you explain this answer].

29 YOUR TURN 6. What is the end behaviour of? y ln x

30 YOUR TURN 7. Is the function y 15log 10 x increasing or decreasing? How do you know?

31 YOUR TURN 8. Sketch a possible graph of a logarithmic function if: a 0 A. B. y a 1 2 y x x

32 YOUR TURN 9. Which graph best represents the function y 2ln x? (A) (B) (C) (D)

33 PG , # 1, 2, 5, 7, 8, 12, 15 Independent practice

34 7.2 EVALUATING LOGARITHMIC Chapter 7 EXPRESSIONS

35 EXPRESSING A LOGARITHMIC EQUATION AS AN EXPONENTIAL EQUATION AND VICE VERSA To evaluate a logarithmic expression, we must write the logarithm in exponential form. Recall log b x y b y x ln x y e y x

36 AN EASY WAY TO REMEMBER! Left Right = Centre log n e b e n b e exponent base exponent argument n the argument b base of log

37 CONVERT THE FOLLOWING TO EXPONENTIAL FORM log log 5 (A) (B) (C) log

38 CONVERT THE FOLLOWING TO EXPONENTIAL FORM log 25 n log (D) (E) (F) ln

39 CONVERT THE FOLLOWING TO LOGARITHMIC FORM: (A) (B) (C)

40 CONVERT THE FOLLOWING TO LOGARITHMIC FORM: (D) 2 (E) y (F) e x x 10 x e y

41 SOLVE FOR X IN THE FOLLOWING LOGARITHMIC EQUATIONS: log 2 64 x log x 4 (A) (B) (C) log x

42 SOLVE FOR X IN THE FOLLOWING LOGARITHMIC EQUATIONS: log 16 4 x ln x 2 (D) (E) (F) log x

43 WHEN THE EQUATION EQUALS X, WE DO NOT NEED AN EQUATION: Evaluate: (A) (B) (C)

44 EVALUATING WITH MORE THAN ONE LOGARITHMIC EXPRESSION Evaluate: (A) log log 2 2 (B) log 2 1 log 2 (1/8)

45 EVALUATING WITH MORE THAN ONE LOGARITHMIC EXPRESSION (C) (log 5 125)(log 7 49)

46 USING YOUR CALCULATOR, EVALUATE THE FOLLOWING. ROUND YOUR ANSWERS TO THE NEAREST HUNDREDTH. log125 log 2500 ln 80 (A) (B) (C)

47 YOUR TURN 1. What is the exponential form of? (A) 3 5 x log 5 3 x (B) (C) (D) 3 x 5 5 x 3 3 x 5

48 YOUR TURN 2. What is the logarithmic form of 2 8? (A) log (B) log (C) log (D) log 6 8 2

49 YOUR TURN 3. Evaluate: log (A) (B) (C) (D)

50 YOUR TURN 4. What is the value of? (A) 1 (B) 2 (C) 5 (D) 6 log 3 5log 1 3 5

51 YOUR TURN 5. Evaluate: (A) log (B) 27 log 3 81

52 YOUR TURN (C) 1 4log 1 81 log

53 PG , #1-4, 6-12 Independent practice

54 7.3 LAWS OF LOGARITHMS Chapter 7

55 LOGARITHM IDENTITIES log b m + log b n = log b (mn)

56 VERIFYING THE LAWS OF LOGARITHMS 1. Verify the Product Law. Show that: 9 27 log 9 log 27 log 3 3 3

57 VERIFYING THE LAWS OF LOGARITHMS 2. Verify the Quotient Law. Show that: 32 log log log 2

58 VERIFYING THE LAWS OF LOGARITHMS 3. Verify the Power Law. Show that: 3 4 3log 4 log 2 2

59 4. WRITE EACH EXPRESSION AS A SINGLE LOGARITHM, AND THEN EVALUATE. (A) log log (B) log log 3 (1.5)

60 4. WRITE EACH EXPRESSION AS A SINGLE LOGARITHM, AND THEN EVALUATE. (C) log (D) log ½log 5 16

61 4. WRITE EACH EXPRESSION AS A SINGLE LOGARITHM, AND THEN EVALUATE. (E) log (F) log 2 3 log 2 48

62 4. WRITE EACH EXPRESSION AS A SINGLE LOGARITHM, AND THEN EVALUATE. (G) log log 5 2 (H) 2log log 2 5

63 5. WRITE EACH OF THE FOLLOWING AS A SINGLE LOGARITHM (A) log6 3 log6 4 log (B) log b A log b M log b Z

64 5. WRITE EACH OF THE FOLLOWING AS A SINGLE LOGARITHM (C) log A log B log C log D (D) 2log3 a log3 b 5log3 c

65 6. EXPRESS LOG20 AS A SUM OF TWO LOGARITHMS.

66 7. EXPRESS LOG20 AS A DIFFERENCE OF TWO LOGARITHMS.

67 REWRITE IN TERMS OF LOGM, LOGN, AND LOGP. (A) log MNP

68 8. REWRITE IN TERMS OF LOGM, LOGN, AND LOGP. (B) log M P 3 2

69 REWRITE IN TERMS OF LOGM, LOGN, AND LOGP. (C) log 3 M N 2 P

70 PG , #1-8, Independent practice

71 7.2/7.3 APPLICATIONS OF Chapter 7 LOGARITHMS

72 MAGNITUDE OF EARTHQUAKES The magnitude of an earthquake is measured on the Richter Scale. Magnitude refers to the amount of energy released during an earthquake. The magnitude of an earthquake, y, can be determined using y=logx, where x is the amplitude of the vibrations measured using a seismograph. An increase of one unit in magnitude results in a 10 fold increase in the amplitude.

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74 (A) DETERMINE THE MAGNITUDE OF AN EARTHQUAKE IF THE AMPLITUDE OF VIBRATIONS, AS MEASURED BY A SEISMOGRAPH, IS 500. y=logx y=log500 y=2.699 This means =500

75 (B) DETERMINE THE MAGNITUDE OF AN EARTHQUAKE IF THE AMPLITUDE OF VIBRATIONS, AS MEASURED BY A SEISMOGRAPH, IS

76 (C) IF THE MAGNITUDE OF AN EARTHQUAKE IS 7.1, DETERMINE THE AMPLITUDE OF VIBRATIONS.

77 PH OF A SOLUTION The ph scale in chemistry is used to measure the acidity of a solution. The ph scale is logarithmic with a base of 10. A logarithmic scale is useful for comparing numbers that vary greatly in size. The ph scale ranges from 0 to 14 with the lower numbers being acidic and the higher numbers being basic. A value where the ph=7 is considered neutral. The scale is a logarithmic scale with one unit of increase in ph resulting in a 10 fold decrease in acidity. Another way to consider this would be a one unit increase in ph results in a 10 fold increase in basicity.

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79 EXAMPLE The ph, p(x), is defined by the equation p(x) = logx where the concentration of hydrogen ions, x, in a solution is measured in moles per litre (mol/l). (A) The hydrogen ion concentration, x, of a solution is mol/l. Calculate the ph of the solution.

80 The ph, p(x), is defined by the equation p(x) = logx where the concentration of hydrogen ions, x, in a solution is measured in moles per litre (mol/l). (B) Lemon juice has a ph of 2. What is the hydrogen ion concentration of lemon juice?

81 The ph, p(x), is defined by the equation p(x) = logx where the concentration of hydrogen ions, x, in a solution is measured in moles per litre (mol/l). (C) In terms of hydrogen ion concentration, how much more acidic is Solution A, with a ph of 1.6, than Solution B, with a ph of 2.5? Round your answer to the nearest tenth.

82 DECIBEL SCALE FOR SOUND LEVELS Sound levels are measured in decibels using the function 10(log I 12) where is the sound level in decibels (db) and I is the sound intensity measured in watts per metre squared (W/m 2 ).

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84 1. WHAT IS THE SOUND LEVEL, TO THE NEAREST DECIBEL, OF EACH SOUND? 11 2 (A) The rustle of leaves, if I 1 10 W / m? 10(log I 12)

85 1. WHAT IS THE SOUND LEVEL, TO THE NEAREST DECIBEL, OF EACH SOUND? (B) Ordinary conversation at 50cm, if? 10(log I 12) I W / m

86 2. THE SOUND LEVEL OF A BASS DRUM IS 110 DB. THE SOUND LEVEL OF A PIANO WHILE PRACTICING IS 60 DB. How many times louder is the bass drum than the piano? Recall than an increase of 10 db represents an increase in loudness by a factor of 10.

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88 PG. 437 # Independent practice

89 7.4 SOLVING EXPONENTIAL EQUATIONS USING LOGARITHMS Chapter 7

90 REMEMBER! Common Base 4 x = (x) =2 7 2x = 7 When we cannot get a common base 4 x =100 (until now we ve used guess and check) x=3.322 x=7/2 or 3.5

91 OR Steps 1. Take the log of both sides of the equation. 2. Use the Power Rule to move the exponent down. 3. Isolate the variable and solve. (Sometimes we will have to use the Distributive Property to simplify first before we solve). Note! When evaluating a logarithm, keep at least 3 decimal places for accuracy.

92 SAME EXAMPLES! 4 x =128 log4 x =log128 4 x =100 log4 x =log100 xlog4=log128 xlog4=log100 x = log128 log4 x = log100 log4 x = 3.5 x = 3.322

93 EXAMPLES 1. Determine the value of y in each exponential equation. Verify your answer. (A) 10 y = 81

94 EXAMPLES 1. Determine the value of y in each exponential equation. Verify your answer. (B) 25 = e y

95 EXAMPLES 1. Determine the value of y in each exponential equation. Verify your answer. (C) 3.2 y = 5/6

96 EXAMPLES 1. Determine the value of y in each exponential equation. Verify your answer. (D) 8 2x =32

97 2. Solve the following exponential equations. Give answer both in exact form and to three decimal places: (A) 3 x 1 = 20

98 2. Solve the following exponential equations. Give answer both in exact form and to three decimal places: (B) 5 2x+3 = 10

99 WHEN WE HAVE EXPONENTS ON BOTH SIDES OF THE EQUATION After we take the log of both sides and move the powers down, unless the question specifies that the answer be in exact form, it is easier to write each log as a decimal (to 3 decimal places) and distribute the resulting number. Then, put like terms together and solve for the variable.

100 EXAMPLES 1. Solve 2 x 1 = 3 x+1, and round your answer to three decimal places.

101 EXAMPLES 2. Solve 5 x+3 = 8 x-2, and round your answer to three decimal places.

102 EXAMPLES 3. Solve 4 2x 3 = 9 -x, and round your answer to three decimal places.

103 SOMETIMES WE WILL HAVE TO ISOLATE THE POWER FIRST Solve for x: x 2( 5) 20 x ( 5) 10 x log 5 log10 log10 x log 5 x xlog 5 log10

104 EXAMPLES. SOLVE FOR X: ( A ) 18 3( 2) x

105 SOLVE FOR X: x ( B ) 15( 6) 7

106 SOLVE FOR X: ( C) x

107 SOLVING LOG EQUATIONS USING A CALCULATOR Recall the examples we did when we evaluated a logarithm by rewriting the expression in exponential form: log (x) 5 log x x x x 2 3 2

108 THIS STRATEGY WORKS FINE WHEN WE CAN GET A COMMON BASE, BUT IT IS NOT USEFUL WHEN WE CANNOT GET A COMMON BASE: log 2 48 x x 2 48 x log 2 log 48 x log 48 log 2 x xlog 2 log 48

109 IN THESE CASES, WE CAN EVALUATE THE LOGARITHM USING A CALCULATOR: New Rule! log b a log log a b

110 EVALUATE THE FOLLOWING: (A) log 2 100

111 EVALUATE THE FOLLOWING: (B) log 4 15

112 EVALUATE THE FOLLOWING: (C) log 1/2 0.8

113 EVALUATE THE FOLLOWING: (D) log 2/3 (5/9)

114 PG , #1, 2, 3, 5, 6, 15, 16 Independent practice

115 7.4 CONTINUED APPLICATIONS OF LOGARITHMIC FUNCTIONS Chapter 7

116 SIMILAR TO UNIT 6, WE WILL BE SOLVING WORD PROBLEMS INVOLVING HALF-LIFE, DOUBLING TIME, INTEREST, ETC Half-Life Problems Remember! In a half-life problem, the amount of a substance decreases by ½ every fixed number of t years. The formula is: 1 h A( t) A0 2 Where: A(t) - the amount of substance present at time t A 0 t h - represents initial amount of the substance present - represents the time - represents the half-life (time for substance to decrease by ½)

117 DOUBLING-TIME PROBLEMS In a doubling time problem, the amount of a substance doubles every fixed number of years. The formula is: t d A t) A 2 ( 0 Where: A(t) - the amount of substance present at time t A 0 - represents initial amount of the substance present t - represents the time d - represents the doubling-life (time for substance to double

118 EXAMPLES: 1.The half-life of a radioactive isotope is 30 hours. The amount of radioactive isotope, at time, can be t modeled by the function A( t) A Determine algebraically how long it will take for a sample of 1000mg to decay to 50mg

119 1 t 2.THE EQUATION 15 A A REPRESENTS THE POPULATION 0 2 OF A TOWN WHOSE HALF -LIFE IS 15 YEARS. Determine the number of years it will take for the population to decrease by 25%.

120 3. THE POPULATION OF TROUT GROWING IN A LAKE CAN BE MODELED BY THE FUNCTION: P( t) 200 t 2 5 Where P(t) represents the number of trout and t represents the time in years after the initial count. How long will it take for there to be 5000 trout?

121 4. SMALL RURAL WATER SYSTEMS ARE OFTEN CONTAMINATED WITH BACTERIA. SUPPOSE A WATER TANK IS INFESTED WITH E. COLI BACTERIA SUCH THAT THE BACTERIA TRIPLE IN NUMBER EVERY 10 DAYS. The number of bacteria present after t days can be modeled by the function: t A( t) How long will it take for there to be bacteria present?

122 COMPOUND INTEREST where: A = future value P = principal A P( 1 i) i = interest rate per compounding period n = number of compounding periods n

123 COMPOUNDING PERIODS CAN BE DAILY, WEEKLY, BI - WEEKLY, SEMI-MONTHLY, MONTHLY, QUARTERLY, SEMI-ANNUALLY, OR ANNUALLY. Compounding Period Daily Weekly Bi-Weekly Semi-Monthly Monthly Quarterly Semi-Annually Annually Number of Times Interest is Paid 365 times per year 52 times per year 26 times per year 24 times per year 12 times per year 4 times per year 2 times per year 1 time per year Interest Rate Per Compounding Period annual rate i 365 annual rate i 52 annual rate i 26 annual rate i 24 annual rate i 12 annual rate i 4 annual rate i 2 annual i 1 rate

124 REMEMBER! Interest Rate 6% Compounded Annually Compounded Quarterly Compounded Monthly

125 1. DEB INVESTS $ I NTO AN A C C O UNT WHICH H AS A C O M P O UND I NTEREST RAT E O F 6%. W RITE AN E X P O NENTIAL E Q UAT I O N REPRESENTING T H E SITUAT I O N A ND USE IT TO DETERMINE H O W LONG IT WILL TA K E FOR HER INVESTMENT TO REACH $5000 I F IT IS COMPOUNDED : (A) Annually.

126 1. DEB INVESTS $ I NTO AN A C C O UNT WHICH H AS A C O M P O UND I NTEREST RAT E O F 6%. W RITE AN E X P O NENTIAL E Q UAT I O N REPRESENTING T H E SITUAT I O N A ND USE IT TO DETERMINE H O W LONG IT WILL TA K E FOR HER INVESTMENT TO REACH $5000 I F IT IS COMPOUNDED : (B) Semi-Annually.

127 1. DEB INVESTS $ I NTO AN A C C O UNT WHICH H AS A C O M P O UND I NTEREST RAT E O F 6%. W RITE AN E X P O NENTIAL E Q UAT I O N REPRESENTING T H E SITUAT I O N A ND USE IT TO DETERMINE H O W LONG IT WILL TA K E FOR HER INVESTMENT TO REACH $5000 I F IT IS COMPOUNDED : (C) Monthly.

128 2. A PRINCIPAL AMOUNT OF $3000 WAS INVESTED IN AN ACCOUNT IN WHICH 8.2% COMPOUND INTEREST WAS COMPOUNDED QUARTERLY. HOW LONG WILL IT TAKE FOR THIS INVESTMENT TO TRIPLE IN VALUE?

129 PG , #8, 10, 11, 12 Independent practice

130

131 7.5 MODELLING DATA USING Chapter 7 LOGARITHMIC FUNCTIONS

132 IN THIS SECTION, STUDENTS WILL: Graph data, and determine the logarithmic function that best approximates the data. Interpret the graph of a logarithmic function that models a situation, and explain the reasoning. Solve, using technology, a contextual problem that involves data that is best represented by graphs of logarithmic functions and explain the reasoning.

133 EXAMPLES 1. What type of curve would best approximate the data in the scatter plot shown? (A) (B) (C) (D) exponential linear logarithmic parabolic

134 2. What function best describes the scatter plot? y x (A) (B) (C) (D) a decreasing exponential function an increasing exponential function a decreasing logarithmic function an increasing logarithmic function

135 3.WHEN OBJECTS OF DIFFERENT MASSES ARE COMPARED WITHOUT A SCALE, THE DIFFERENCE IN MASS IS NOT ALWAYS PERCEIVED. THE MINIMUM PERCEIVABLE DIFFERENCE FOR VARIOUS MASSES IS SHOWN. Mass (g) Minimum Perceivable Difference (g)

136 THIS DATA CAN BE MODELLED USING LOGARITHMIC REGRESSION. (A)Write the equation for the logarithmic regression function. Round values of a and b to the nearest tenth.

137 (B) BASED ON THE REGRESSION EQUATION, DETERMINE THE MINIMUM PERCEIVABLE DIFFERENCE FOR A 2100G OBJECT, TO THE NE AREST WHOLE GRAM.

138 4. The flash on most digital cameras requires a charged capacitor in order to operate. The percent charge, Q, remaining on a capacitor was recorded at different times, t, after the EXAMPLE flash had gone off. 4 The t.5 flash duration represents the time until a capacitor has only 50% of its initial charge. The t.5 flash duration also represents the length of time that the flash is effective, to ensure that the object being photographed is properly lit. a) Construct a scatter plot for the given data. y x

139 The flash on most digital cameras requires a charged capacitor in order to operate. The percent charge, Q, remaining on a capacitor was recorded at different times, t, after the flash had gone off. The t.5 flash duration represent the time until a capacitor has only 50% of its initial charge. The t.5 flash duration also represents the length of time that the flash is effective, to ensure that the object being photographed is properly lit. b) Determine a logarithmic model for the data. LnReg y=a+blnx a= b=

140 The flash on most digital cameras requires a charged capacitor in order to operate. The percent charge, Q, remaining on a capacitor was recorded at different times, t, after the flash had gone off. The t.5 flash duration represent the time until a capacitor has only 50% of its initial charge. The t.5 flash duration also represents the length of time that the flash is effective, to ensure that the object being photographed is properly lit. c) Use your logarithmic model to determine the t.5 flash duration to the nearest hundredth of a second. y x

141 PG , #2, 3, 6, 7, 9, 11. Independent practice

142 EXAMPLE Yvonne has a balance of $3215 in her savings account. This account pays 2.4% interest per year, compounded annually. The compound interest formula is: A = P(1 + i) n Where A represents the future value, P represent the principal, I represents the interest rate applied each compounding period, and n represents the number of compounding periods. How long will it take for Yvonne s balance to reach $5000?

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144 EXAMPLE Caffeine is found in coffee, tea, and soft drinks. Many people find that caffeine makes it difficult for them to sleep. The following data was collected in a study to determine how quickly the human body metabolizes caffeine. Each person started with 200 mg of caffeine in her/his bloodstream, and the caffeine level was measured at various times. Enter the data on the handout into your calculator. a) Determine the equation of the logarithm regression function for the data representing time as a function of caffeine level. b) Determine the time it takes for an average person to metabolize 50% of the caffeine in his/her bloodstream. Round to the nearest tenth of an hour. c) Paula drank a cup of coffee that contained 200 mg of caffeine at 10:00 am. How much caffeine will be in her bloodstream at 9:00 pm? Round your answer to the nearest milligram.