Chapter 8. Exponential and Logarithmic Functions

Transcription

1 Chapter 8 Eponential and Logarithmic Functions

2 Lesson 8-1 Eploring Eponential Models

3 Eponential Function The general form of an eponential function is y = ab. Growth Factor When the value of b is greater than 1 Decay Factor When the value of b is less than 1 When you see words like Increase or appreciation, think growth Decrease or depreciation, think decay

4 Eample 4 Page 427, #16, 20 Without graphing determine whether each function represents eponential growth or eponential decay. y 129(1.63) y Eponential Growth Eponential Decay

5 Eample 1 Page 426, #4 Graph each Function. y 9(3) Eponential Growth y

6 Eample 5 Page 427, #26 Graph each Function. Eponential Decay y 0.25 y

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

8 Eample 2 A Carl s weight at 12 yr is 82 lb. Assume that his weight increases at a rate of 16% each year. Write an eponential function to model the increase. Calculate his weight after 5 yr. Step 1 Find a and b. y ab a is the original a 82 b is the growth factor or decay factor b 1r

9 Eample 2 A Step 2 Find the eponential function. y ab y 82(1.16) a b Step 3 Calculate his weight after 5 yr. 5 y 82(1.16) If the model is correct, Carl will weight about 172 lb in 5 yr.

10 Eample 2 B A motorcycle purchased for $9000 today it will be worth 6% less each year. For what can you epect to sell the motorcycle at the end of 5 yr? Step 1 Find a and b. y ab a is the original a 9000 b is the growth factor or decay factor b 1r

11 Eample 2 B Step 2 Find the eponential function. y ab y 9000(0.94) a b Step 3 Calculate the sale price after 5 yr. 5 y 9000(0.94) $

12 Lesson 8-2 Properties of Eponential Functions

13 Eample 1 Page 434, #2 Graph each function. Label the asymptote of each graph y 1 2 y asymptote is y = 0

14 Eample 3 Page 434, #16 Iodine-131 is used to find leaks in water pipes. It has a half-life of 8.14 days. Write the eponential decay function for a 200-mg sample. Find the amount of iodine-131 remaining after 72 days. Step 1 Find a and b. y ab a is the original a 200 b is ½ and is the number of days 1 b

15 Eample 3 Page 434, #16 Step 2 Find the eponential function. y ab y a b Step 3 Calculate the amount after 72 days. y mg

16 Graph of y = e

17 Eample 4 Page 434, #18 Use the graph of y = e to evaluate each epression to four decimal places. Use your calculator too. 3 y e

18 Continuously Compound Interest Formula Amount in account A Pe Principal Annual rate of interest rt Time in years

19 Eample 5 Page 434, #24 Find the amount in a continuously compounded account for the given conditions. principal: $2000 annual interest: 5.1% time: 3 yr A Pe rt 0.051(3) 2000e $

20 Eample A Suppose you invested $1050 at an annual rate of 5.5% compound continuously. How much you will have in the account after 5 years. Step 1 Find A, P, r and t A P r t?

21 Eample A Step 2 Find A A Pe rt 1050e $ (0.055)(5) A P r t?

22 Lesson 8-3 Logarithmic Functions as Inverses

23 Logarithmic Function y 10 y log

24 Logarithm The logarithm to the base b of a positive number y is defined as follows: y b log y b

25 Eample 2 Page 442, #6 Write the equation in logarithmic form y b log b y log7 49 2

26 Eample 2 Page 442, #12 Write the equation in logarithmic form y b log y b log 3 27

27 Eample 3 Page 442, #14 Evaluate the logarithm log2 16 log216 log b y y b

28 Eample 3 Page 442, #15 Evaluate the logarithm log4 2 log4 2 log b y y b

29 Common Logarithm A common logarithm is a logarithm that use base 10. log10 y log y

30 Eample Page 442, #46 Use your calculator to evaluate the logarithm to four decimals places. Then find the largest integer that is less than the value of the logarithm. log Largest integer is

31 Eample Page 442, #58 Write the equation in eponential form. 1 log log y y b b

32 Lesson 8-4 Properties of Logarithms

33 Properties of Logarithms Product Property log MN log M log N b b b Quotient Property M log log M log N b b b N Power Property log b M log M b

34 Eample 1 Page 449, #2, 4, and 8 State the property or properties used to write each epression. #2) log3 32 log3 8 log3 4 Quotient Property #4) log n p p 6 log6 n Power Property #8) 2 4 2logw 4logz logw z Power Property Product Property

35 Eample 2, Page 449, #14 Write each logarithmic epression as a single logarithm log8 2log6 log3 log8 log6 log3 log8 log36 log3 8 log log log log3 9 2

36 Eample 2, Page 449, #14 2 log 9 2 log log 3 log3

37 Eample 2, Page 449, #18 Write each logarithmic epression as a single logarithm log log y log z log y log 7 7 z log y 7 z

38 Eample 3 Page 449, #22 Epand the logarithm 4 2 log3m n log m n log3m 4 2 logn 4 2 log3 logm logn log3 4logm 2logn

39 Eample 3 Page 449, #28 Epand the logarithm log 8 3a 8 5 log 8 log 3a log log 3 a log 8 log 3 log a

40 Eample 3 Page 449, #28 log 8 log 3 log a log 8 log 3 log a log 3 log a

41 Eample Page 449, #34 Use properties of logarithms to evaluate each epression. 3log 2 log () 1 log log ( 1)

42 Eample Page 449, #38 Use the properties of logarithms to evaluate the epression. 1 2log84 log log 4 log 8 log 8 log log

43 Lesson 8-5, Part 1 Eponential and Logarithmic Equations

44 Eponential Equation Eponential Function Eponential Equation y b b c a

45 Eample 1 Page 456, #2 Solve the equation. Round your answer 4 decimal places. log log19 log

46 Eample 1 Page 456, #10 Solve the equation. Round your answer 4 decimal places log log144 log log log25 log144 1 log

47 Change of Base To evaluate a logarithm with any base, you can use the Change of Base Formula log b M logm logb

48 Eample 2 Page 456, #16 Use the Change of base formula to evaluate the epression. Then convert it to a logarithm in base 8 log 7 log7 2 log2 log 7 log log

49 Eample 2 Page 456, # log 7 is approimately equal to or log

50 Eample 3 Page 456, #20 Use the Change of Base Formula to solve the equation log log21 2 log6 log21 log

51 Eample 3 Page 456, #26 Use the Change of Base Formula to solve the equation log 4 log ( 2) log 4 log (1) log 89 2 log log89 log4 log89 log

52 Lesson 8-5, Part 2 Eponential and Logarithmic Equations

53 Eample 6 Page 456, #34 Solve the equation. Check your answers. 2log log log

54 Eample 6 Page 456, #40 Solve the equation. Check your answers. 2log( 1) 5 2log( 1) log( 1)

55 Eample 7 Page 457, #42 Solve the equation. log log3 8 log

56 Eample 7 Page 457, #46 Solve the equation. 3log log6 log log log6 log log log log 9 6

57 Eample 7 Page 457, # log 9 6 log

58 Lesson 8-6 Natural Logs

59 Natural Logarithmic Function If y e, then log, which is commonly written as ln y. e y e y e ln y

60 Properties of Natural Logarithms Product Property lnmn lnm lnn Quotient Property M ln lnm ln N N Power Property lnm lnm

61 Eample 1 Page 464, #4 Write the epression as a single natural logarithm. 4ln8 ln10 4 ln8 ln10 4 ln8 (10) ln40960

62 Eample 1 Page 464, #8 Write the epression as a single natural logarithm. 1 1 ln ln y 4ln z ln 4ln 3 3 y z ln y 1 3 lnz ln y z 4

63 Eample 1 Page 464, #8 1 3 ln y z 4 ln 3 z y 4

64 Eample 2 Page 464, #10 Find the value of y for the given value of y 15 3ln, for 7.2 y y 15 3ln

65 Eample 3 Page 465, #14 ln y Solve each equation ln3 6 3 e e 3 3 e y e

66 Eample 3 Page 465, #20 ln y Solve each equation y e 1 ln e e 2 e

67 Eample 4 Page 465, #24 Use natural logarithms to solve each equation. e 2 10 ln10 2 ln y y e ln

68 Eample 4 Page 465, #28 Use natural logarithms to solve each equation. 9 e 86 e (ln14) ln y y e e 9 ln

69 Eample Page 466, #62 Solve each equation ln( 2) ln e 4 2 ln e 3 e