1

Honors Algebra 2 ~ Spring 2014 Unit 6 ~ Chapter 8 Name______________________

Unit 6: Exponential & Logarithmic Functions

NC Objectives: 1.01 Simplify and perform operations with rational exponents and logarithms to solve problems.

2.01 Us the inverse of functions to model and solve problems.

2.03 Use exponential functions to model and solve problems.

a. Solve using tables, graphs, and algebraic properties.

b. Interpret the constants, coefficients, and bases in the context of the problem.

2.04 Create and use best-fit mathematical models of exponential functions to solve problems involving sets of data.

DAY DATE LESSON ASSIGNMENT

1 Mon.

April 21 Exponential Models and Best Fit Packet p. 2 & 3

2 Tues.

April 22

Section 8.2: Properties of Exponential

Functions

Half-life, compound & continuous interest

Packet p. 4(all)

Packet p. 5 #8-14

3

Wed.

April 23

Quiz on Sections 8.1-8.2

Section 8.3: Logs & Logarithmic Functions

as Inverses

Packet p. 5 #1-21

Packet p. 6 #30-57

Watch video: http://patrickjmt.com/properties-

of-logarithms/

4 Thurs.

April 24 Section 8.4: Properties of Logarithms

Properties of Logs Handout

ODD Problems

5 Fri.

April 25

Finish Properties

(LOG 21) Packet p. 6 #1-18

6 Mon.

April 28

Review Sections 8.3 & 8.4

Packet Page 7

Study for Quiz on 8.3-8.4

7 Tues.

April 29

QUIZ on Sections 8.3-8.4

Section 8.5: Solving Exponential & Log

Equations

Packet Page 8: Part A

8 Wed.

April 30

Finish Section 8.5

Packet Page 8: Part B

Packet page 9

9 Thurs.

May 1

Section 8.6: Natural Logarithms

Application Problems Using Logs Packet p. 10

10 Fri.

May 2 Application of Logs/Review Handout

11 Mon.

May 5

Review for Unit 6 Test

Test is calculator active and inactive. Packet p. 11 STUDY FOR TEST

12 Mon.

May 6

Unit TEST 6 (Chapter 8)

Calculator Active and Inactive test

Packet p. 13 & 14

Print notes & packet for unit 7

Chapter 9

2

Homework Section 8.1

I. Identify each function as modeling either exponential growth or exponential decay. What is the function’s percent increase or decrease? 1. y = 1298(1.63)x 2. y = 0.65(1.3)x 3. y = 2(0.65)x 4. 12(1.7)x

5. y = 5(6)x 6. y =

5(0.45)

6x

7. On their federal income tax returns, many self-employed individuals can depreciate the

value of the business equipment they purchase. Suppose a computer valued at $6500 depreciates at a rate of 14.3% per year. After how many years is the value of the computer less than $2000?

II. Write an exponential function to model each situation. Find the value of the function after

five years. 8. A population of 250 frogs increases at an annual rate of 22%. 9. A stock priced at $35 increases at a rate of 7.5% per year. 10. A $17,500 delivery van depreciates 11% each year. 11. A population of 115 cougars decreases 1.25% each year.

III. Complete the following.

12. Todd says y = x2 is an exponential function. Juan disagrees. Who is correct? Explain. 13. Solve the following equations using the graphing calculator.

a. 3 = 2.3x b. 0.9x = 5

14. A herd of Tule Elk was introduced into the Point Reyes National Seashore in 1978. The number of Elk present is a function of time.

Year 78 79 80 81 82 83 84 85 86 87 88 89

Pop. 10 15 24 32 41 55 70 82 96 109 132 169

a. Write a prediction equation to show how the number of Elk and the year are related. b. Predict the number of elk in 1995. c. Estimate when there will be 2771 Elk

15. A tractor that 6 years ago cost $10,000, is now worth only $3200. Find the average

annual rate of depreciation.

3

Modeling Homework The data in the table give the average speed y (in knots) of the Trident motor yacht for several different engine

speeds x (in hundreds of revolutions per minute or RPMs).

Engine Speed

(in 100’s)

9

11

13

15

17

19

21.5

Boat Speed

6.15

7.23

8.76

9.06

11.01

12.43

14.98

1. a.) Find the linear model. Round to 4 decimal places.

b.) What is the R2 value? _______________

2. a.) Find the quadratic model. Round to 4 decimal places.

b.) What is the R2 value? _______________

3. a.) Find the cubic model. Round to 4 decimal places.

b.) What is the R2 value? _______________

4. a.) Find the exponential model. Round to 4 decimal places.

b.) What is the R2 value? _______________

5. Which model is the best?

6. Use the best model to estimate the speed of the Trident for an engine speed of 2400 RPMs. _____________

***************************************************************************************************

7. Find a linear, quadratic, and cubic model for the data. Use the model that best fits the data to estimate the

diving record in 2004.

Men's Olympic Springboard Diving Records

Year 1980 1984 1988 1992 1996 2000

Points 905.02 754.41 730.80 676.53 701.46 708.72

8. The best temperature to brew coffee is between 195˚F and 205˚F. Coffee is cool enough to drink at 185˚F.

The table shows temperature readings from a sample cup of coffee. How long does it take for a cup of coffee

to be cool enough to drink? Use an exponential model.

Time(min) 0 5 10 15 20 25 30

Temp(F˚) 203 177 153 137 121 111 104

4

Homework Day 2: Half-life, compound and continuous interest

Part 2

5

6

Use 5log 2 0.4307 and 5log 3 0.6826 to evaluate each expression.

Solve each equation.

5 5

5 5

5 5

21)log 9 2) log

3

3) log 50 4) log 30

105) log 6) log .5

9

6 6 6 8 8 8

7 7 3 3 3

9 9 9

7)log log 9 log 54 8) log 48 log log 4

2 1 19) log log 8 10) log log 16 log 64

3 4 3

11) log (3 14) log 5 log 2u

x w

n y

u

7 7 7 7

2 2 2 6 6

216 16 8 8

12) log log log 3 log 12

13) 4log log 5 log 405 14) log (2 5) 1 log (7 10)

115) log (9 5) log ( 1) 16) log ( 3) log ( 4) 1

2

17)

x x

x x x

x x n n

6 6 6 6

22 2

log (3 7) log ( 4) 2log 6 -3log 3

18) log (2 8) log (2 21 61) 3

m m

x x x

7

Review 8.3 & 8.4 LOGARITHMS!!!

Evaluate:

1. 81log3 2. 0001.0log 3. 16

1log2 4. 5log25 5. 27log

3

1

6. 1log9 7. 4log8 8. 37log57log7 9. ))9(log(loglog 3210

Expand using the properties of Logs:

10. 27 3log x 11.

36logbc

a

Write as a single logarithm with a coefficient of 1.

12. cba 222 log5log3log 13. )log2log3(7

155 yx

Evaluate:

14. 53log23 15. 7log aa 16.

35log225log35 17.

122log62log22

18. 53log

9 19. 32log

8

Given: 585.13log 2 and 322.25log2 find the following:

20. 180log2 21. 45.0log2 22. 25

6log2

Solve.

23. 216log x 24. 3

2log8 x 25. x25log

5

26. 2

13log x 27. x8log16 28. 2log37log2log 555 x

29. 2)21log(log xx 30. 1)1log()2log( xx 31. 416log x

32. 6

19log 3 x 33. xx 5log)6(log 4

24 34. 2log6 x

35. x5.log2 36. 4

13log x 37. 2)4(log3 x

38. )2log24log2(6

1log x 39. x

8

1log

2 40. )5(log1)1(log 77 xx

8

Section 8.5 Homework: Solving Exponential Equations(PART:A)

Use logarithms to solve each equation.

Section 8.5-8.6 Homework(PART B)

Use logarithms to solve each equation. Round to three decimal places.

1. 9b = 45 2. 3.1a-3 = 9.42 3. 3.53x+1=65.4 4. 55a-2 = 22a+1 5. 5x-1=3x 6. 53y = 8y-1 Solve each equation 7. ex = 18 8. e2x = 10 9. ln (4x – 1) = 36

10. ex+1 = 30 11. 5 4 7x

e 12. ln (3x + 5) = 4

13. The function U(t) = 1.31e.548t describes how the number of internet users, in millions, increased

exponentially from 1990 to 1995. Let t represent the time, in years, since 1990.

a. What was the first year in which there were 13 million internet users?

b. How many years did it take for the number of users to double since 1990?

c. Solve the equation U = 1.31e.548t for t.

14. Suppose $500 is invested at 6% annual interest compounded twice a year. When will the

investment be worth $1000?

15. An organism of a certain type can grow from 30 to 195 organisms in 5 hours. Find k for the growth

formula. Use: kty ne .

16. A piece of machinery valued at $250,000 depreciates at 12% per year by the fixed rate method.

After how many years will the value have depreciated to $100,000?

17. A substance decomposes radioactively. Its half-life is 32 years. Find the constant k in the decay

formula. Use: kty ne .

y 4 b+1

b+13

1) 3.5 47.9 2) 8.2 64.5 3) 37.2 8.21 4) 2 7.31

4) 2 7.31 5) y log 78.5

x a

2

4

22 1 3b 2 3

93 3x 18

6) k log 91.8

7) 4 9 8) 7 12 9) 17c 44

10) 7x 111 11) 5 72 12) 4 8

x x b

x x

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More Practice

FORMULAS:

Compound Interest : tn

n

rPA

1 Continuously Compounded : trPeA

“n times a year “

Value of an Asset : (1 )tV P r Growth &/ Decay : tkney

1.) How long would it take for an investment of $2500 to triple if it is invested in an

account that earns 6% interest compounded quarterly.

2.) Your bank promises to double your money in 2

18 years. Assuming the interest rate is

compounded continuously, what is the interest rate?

3.) Zeller industries bought some equipment for $50,000. It is expected to depreciate

at a steady rate of 10% a year. When will the value be half the original value?

4.) The Jameson's bought a new house for $144,500 five years ago. The home is now

worth $187,850. Assuming a steady rate of growth, what was the yearly rate of

appreciation?

5) You have inherited land that was purchased for $30,000 in 1960. The value of the land

increased by approximately 5% per year.

a) Write a model for the value of the land t years after 1960.

b) What is the approximate value of the land in the year 2010?

c) At what year would the land be valued at about half a million dollars?

6) Dave bought a new car 8 years ago for $8400. To buy a new car comparably equipped

now would cost $15,500. Assuming a steady rate of increase, what was the yearly rate

of inflation in car prices over the 8-year period?

7) An investment service promises to triple your money in 12 years. Assuming continuous

compounding of interest, what rate of interest is needed?

10

Homework Day 9

Solve each equation. Round answers to the nearest hundredth.

1. 2log3 log9 1 2. log log4 1 3. log log4 2

3. log log4 3 4. 2log log4 2 5. log(2 5) 3

6. 2log(2

x x x

x x x

x 5) 4 7. log4 1 8. 2log log3 1x x

Use natural logarithms to solve each equation. Round answers to the nearest hundredth.

6 3 2 3 1

3 5 ln2

1. 5 0.1 2. 50 3. 4 5

4. 3 49 5. 25 6. 21

x x x

xx x

e e e

e e e

Solve each equation. Check your answer. Round answers to the nearest hundredth.

2

1. 2ln(3 4) 7 2. 3 4ln(8 1) 7 3. ln ln4 2

24. 3ln 12 5. 5ln 3 2 15 6. ln 2

41x

x x x

xe x

Write each expression as a natural logarithm.

1. ln16 -ln8 2. 3ln3 ln9 3. aln4 ln

4. ln 3ln 5. 5ln ln3 6. 4ln 3ln

b

z x x x x y

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Unit 6 ~ Chapter 8 Review

1. A population of 250 frogs increases at an annual rate of 22%. How many frogs are there after 5

years?

2. A population of 115 cougars decreases 1.25% each year. How many cougars are left after 7 years?

3. Classify each function as modeling growth or decay. Then find the functions percent increase or

decrease.

a. ( ) 3(1.82) xf x b. 1

( ) (.42)3

xf x

4. Graph the function: 1

( )4

x

f x

5. Write an equation in the form xy ab with base 5, passing through (2, 30).

6. Troy’s parents started a savings account for him when he was born. They invested $200 in an

account with 8% annual interest compounded quarterly. How much is in the account on his 18th

birthday?

7. Suppose you invest $3050 at an annual interest rate of 4.5% compounded continuously. Find the

amount in the account after 6 years.

8. Arsenic-74 is used to locate brain tumors. It has a half-life of 17.5 days. Write the exponential decay

function of a 90-mg sample. Use the function to find the amount remaining after 8 days.

9. Write the equation 4

327 81 in logarithmic

form.

10. Write the equation 67log 4489 2 in

exponential form.

11. Evaluate: 2

1log

8.

12. Evaluate: 2

1log

16.

13. Evaluate: 7log 343 .

14. Graph: 2log ( 2) y x

15. Graph: 3log ( 1) y x

16. Given 6log 12 1.387 and 6log 8 1.161

find 6log 96 .

17. Expand 2log 2( 1)x

18. Expand 2

3log3

19. Solve 4

33 72x

20. Solve 7.12 89x

21. Solve 316 92x

22. Solve 3 3 3log 81 log 9 log x

23. Solve 4log ( 6) 2 x

24. Solve ln(2 7) 12 x

25. Solve 2

9 7.2 14.9 x

e

26. Solve 4 4 4log log 3 2log 5 x

27. Solve 3 53 9 x

28. Solve 4 78 3 x x

29. Use the change of base formula to find

3log 198

30. The number of people in the Denver

suburban area has grown exponentially

since 1950 according to the equation

0 ktP Pe . In 1950 there were 350,000

people and in 1960, there were 705,300.

How many people lived in the Denver

suburban area in 1980?

12

Review Answers

Unit 6 ~ Chapter 8 Review - Answers

1. 676 11.

12 ; 3

8

x x 21. 0.544x

2. 105 12. 4x 22. 9x

3. a) growth; 82%

b) decay; 58%

13. 3 23. 6.0625x

4.

14.

24. 81373.89x

5. 6

(5)5

xy 15.

25. 13.930x

6. $832.23 16. 2.548 26. 75x

7. $3995.39 17. log2 2log( 1)x 27. 1x

8. 65.559 mg 18. log8 log27 28. .6397x

9. 27

4log 81

3

19. 10.843x 29. 4.814x

10. 267 4489 20. 2.287x 30. 0.07006

Pop. in 1980 = 2,864,082

k

13

Cumulative Review!!!

1.) Simplify 32272183

2.) In 1984, the population in Greensboro, N.C. was 197,910. Since then it has been growing at the

Rate of 6.9% annually. What equation models the population t years after 1984?

A. 197,910(1 .06)ty B. 197,910(1 69)ty

C. 197,910(1 6.9)ty D. 197,910(1 .069)ty

3.) Solve 2

33 3 4 52x

4.) Solve : 435 xx

A. {4} B.

1,4

1 C. {-1, 4 } D.

5.) If f(x) = -|x| and g(x) = 3x2, what is f(g(-3))?

A. –27 B. –3 C. 3 D. 27

6.) Simplify: 2

1

2

1

ny

y

7.) Exactly 79 feet of wallpaper border were used to decorate around the ceiling of a rectangular

room. To the nearest square foot, what is the maximum possible area of the room?

A. 20 ft2 B. 40 ft2 C. 195 ft2 D. 390 ft2

8.) Using xy 2

8

1 , what is x when y = 8?

A. 4 B. 6 C. 8 D. 10

9.) Ellen deposited $300 in an account that pays 6% interest compounded continuously. How long

will it take for her money to triple?

A. 7.95 years B. 11.55 years C. 18.31 years D. 23.10 years

10.) Solve 2

ln54 xex

14

11.) Three years ago a house was bought for $92,000. Today it is worth $113,000. Find the

approximate yearly rate of appreciation.

A. 7.1% B. 13.6% C. 22.8% D. 61%

12.) Simplify: 2( 3)

(3 9)x

xx

A. x

x3

3 B.

3

( 3)x x C.

3( 3)x

x

D.

2

2

( 3)

3 9

x

x x

13.) Find an exponential function of the form y = abx whose graph passes through (2, 6) and (3, 8).

14.) Solve 131 5.46 xx

15.) Relative to the graph of y = 2x2, how could the graph of y = 2(x-1)2 + 2 best be described?

A. It shifts 1 unit to the right and 2 units down.

B. It shifts 1 unit to the right and 2 units up.

C. It shifts 1 unit to the left and 2 units down.

D. It shifts 1 unit to the left and 2 units up.

16.) Let f(x) = x2 – 3x + 1 and g(x) = 4x – 2 . What is g(f(x))?

A. 4x2 – 12x + 2 B. 4x2 – 12x + 4 C. 16x2 – 28x + 3 D. 16x2 – 28x + 11

17.) Divide x2 – x – 8 by x – 2 .

A. 2

61

xx B.

2

23

xx C.

2

42

xx D.

2

122

xx

18.) Solve: x2 – 2y2 = 8

2x2 + y2 = 36

A. {(4,2). (4,-2), (-4,2), (-4,-2)} B. {(0,6), (0,-6)}

C. {(2,4). (2,-4), (-2,4), (-2,-4)} D. {(2 2,0), ( 2 2,0)}

19.) Which equation best fits the data in the given table? # half-lives 0 1 2 3 4 5 6

Remaining

amount

4,000 2,000 1,000 500 250 125 62.5

A. x

y

2

1000,4 B.

x

y

2

1000,2 C.

1(4000)

2

xy D. 1

(2000)2

xy

20.) Solve 20314 x .

A. 3

1112 x B. 2

3

111 x C.

3

1112 xorx D. 2

3

111 xorx