�1

U6 - Logs and Exponents Name ________________________

Exponents

- Exponents and _______________ go hand in hand. How do you get rid of a square root? _______________ it. How do you get rid of a square? _______________ it.

- Because of this relationship, we can write any root as an _______________ .

Solve.

a. � b. � c. � d. �

e. � f. �

- Also, you can use this relationship to solve equations involving _______________ .

Solve the equation.

g. � h. � i. � j. �

k. � l. �

- There is also a ____________________ Property that allows you to work with just the exponents if the _______________ are the same.

Use One-to-One Property to solve problems for x.

a. � b. � c. � d. �

932 32

−25 814 − 49

12⎛

⎝⎜⎞⎠⎟

−83( )5 312525

2x4 = 162 x − 2( )3 = 10 x3 = 125 3x5 = −3

x + 4( )2 = 0 x4 − 7 = 9993

9 = 3x+1 12

⎛⎝⎜

⎞⎠⎟x

= 8 8 = 22x−1 13

⎛⎝⎜

⎞⎠⎟− x

= 27

�2Logarithms

- _______________ are the inverse of exponential problems. Therefore, they share all the same properties.

Evaluate each expression.

a. � b. � c. � d. �

e. � f. � g. � h. �

i. � j. � k. �

Use One-to-One Property to solve problems for x.

a. � b. � c. �

d. �

Natural Log

- There are two types of exponential functions. You have your _______________ exponential function, and you have your _______________ exponential function. This comes from the fact that the constant e is called the _______________ _______________ .

log3 81 log5 0.04 log12

8 log9 3

log2 32 log31 log4 2 log101100

log61 log51125

log1010,000

log3 x = log312 log 2x +1( ) = log3x log4 x2 − 6( ) = log4 10

log5 x2 + 3( ) = log512

�3- And because logs are simply the _______________ of exponentials, this means there are two log functions

too. You have your _______________ log function (Base _____ ), and you have your _______________ log function (Base _____ ). Even though they share different bases, they are still both log functions and, therefore, share the same log _______________ .

Simplify each expression.

a. � b. � c. � d. �

e. � f. � g. � h. �

Practice

- Rewrite each root as an exponent.

1. � 2. � 3. � 4. � 5. �

- Rewrite each exponent as a root.

6. � 7. � 8. � 9. � 10. �

- Evaluate each expression without a calculator.

11. � 12. � 13. � 14. �

ln 1e

eln5 ln13

2 lne

lne13 5 ln1 3

4lne eln 7

144 113 57( )2 169( )5 28( )11

613 7

14 10

37 5

25 8

74

643 −10003 − 646 4−12

�4

15. � 16. � 17. � 18. �

19. � 20. � 21. � 22. �

- Use One-to-One Property to solve problems for x.

23. � 24. � 25. � 26. �

- Evaluate each logarithm at the indicated value of x.

27. � 28. � 29. � 30. �

31. � 32. � 33. � 34. �

35. � 36. � 37. � 38. �

39. � 40. � 41. � 42. �

43. � 44. �

113 − 256

14⎛

⎝⎜⎞⎠⎟

164( )2 −273( )−4

06( )3 − 25− 32⎛

⎝⎜⎞⎠⎟

3245 −125( )−

23

3x+1 = 27 2x−3 = 16 12

⎛⎝⎜

⎞⎠⎟x

= 32 5x−2 = 1125

log2 64 log25 5 log81 log10

loga a2 logb b

−3 10log4 log5125

log7 343 log81 log1212 log6 36

log4 16 log9 729 log7 2401 log14

14

log412

log15

25

�5- Use One-to-One Property to solve problems for x.

45. � 46. �

47. � 48. �

Solve for x.

49. � 50. � 51. � 52. �

53. � 54. � 55. � 56. �

57. � 58. � 59. � 60. �

Condensing Logs

- With exponents, when you _______________ like bases you add the exponents. When you _______________ like bases you subtract the exponents. Also, when you raise a power to a power you _______________ . There are similar properties with logs that allow you to _______________ or _______________ a log expression based on what is going on with the bases.

log5 x +1( ) = log5 6 log2 x − 3( ) = log2 9

log 2x +1( ) = log15 log 5x + 3( ) = log12

2x = 32 ln x − ln 3= 0 13

⎛⎝⎜

⎞⎠⎟x

= 9 ex = 7

ln x = −3 log x = −1 log3 x = 4 2x = 512

log6 x = 3 5 − ex = 0 9x = 13

3 2x( ) = 42

�6- Use properties of logs to condense each expression.

a. � b. �

Condense each logarithmic expression.

1. � 2. � 3. � 4. �

5. � 6. � 7. � 8. �

9. � 10. �

11. � 12. �

13. � 14. �

log2 7 + 3log2 x − log2 y log6 + 2 log2 − log3

ln2x + ln x log5 8 − log5 t 2 log2 x + 4 log2 y23log7 z − 2( )

ln 4 + ln x log310 + log3 x 4 log8 x log10 y − log10 2

ln x − ln x +1( )+ ln x −1( )⎡⎣ ⎤⎦ 4 ln z + ln z + 5( )⎡⎣ ⎤⎦ − 2 ln z − 5( )

2 ln x + 12ln y − ln z( ) 4 log2 x +

12log2 y − 3log2 z( )

2 log5 x − 2 log5 y − 3log5 z log10 x + 4 log10 y − 5 log10 z

�7

15. � 16. �

17. � 18. �

Exponential/Log Word Problems

- Your basic exponential function is in the form _______________ . Where a is your _______________ value and b is your _______________ . Your basic function _______________ or _______________ depending on if your multiplier is __________ or __________ respectively.

- In the problems that we will be working with, if � then we are dealing with exponential _______________ . But if � , we are dealing with exponential _______________ .

- Remember when you have to write the equation of an exponential equation with only

___________________ you can use � to help find your multiplier.

- When dealing with percentages you can use the _______________ factor or the _______________ factor to help find your multiplier.

- Your growth factor is _______________ and your decay factor is _______________ , where r is your percent of increase or decrease written as a decimal.

- Use exponential equations to solve the following word problems.

a. In January, 1993, there were about 1,313,000 Internet hosts. During the next five years, the number of hosts increased by about 100% per year.

• Write an equation giving the number h (in millions) of hosts t years after 1993.

• About how many hosts were there in 1996?

• When where there 30 million hosts?

143ln x + ln x2 + 3( )⎡⎣ ⎤⎦

122 ln x + ln x + 2( )⎡⎣ ⎤⎦

13log8 y + 2 log8 y + 4( )⎡⎣ ⎤⎦ − log8 y −1( ) 1

2log4 x +1( )+ 2 log4 x −1( )⎡⎣ ⎤⎦ + 6 log4 x

b >10 < b <1

bx2−x1 = y2y1

�8- If you’re working with _______________ _______________ you have to be aware of the annual interest

and how often it is compounded.

b. You deposit $1000 in an account that pays 8% annual interest. Find the balance after 1 year if the interest is compounded with the given frequency.

• annually

• quarterly

• daily

c. You buy a new car for $24,000. The value y of the car decreases by 16% each year.

• Write an equation for the for the value of the car and find the value after 2 years.

• When will the car have a value of $12,000?

d. You buy a new computer for $2100. The value of the computer decreases by about 50% annually.

• Write an equation for the value of the computer and find its value after 2 years.

• When will the computer have a value of $600.

Practice

1. A 100‑gram sample of a radioactive isotope decays at a rate of 6% every week.  How big will the sample be one year from now? 

2. The math club is fast becoming one of the most popular clubs on campus because of the fabulous activities it sponsors annually for Pi Day on March 14.  Each year, the club’s enrollment increases by 30%.  If the club has 45 members this year, how many members should it expect to have 5 years from now?   

�93. Barbara made a bad investment.  Rather than earning interest, her money is decreasing in value by 11%

each week!  After just one week, she is down to just $142.40.  How much money did she start with?  If she does not withdraw her money, how long will it be before she has less than half of what she originally invested?  

4. Larry loves music.  He bought $285 worth of MP3 files on his credit card, and now he cannot afford to pay off his debt.  If the credit-card company charges him 18% annual interest compounded monthly, how does Larry’s debt grow as time passes?  How much would he owe at the end of the year if he had a “no payments for 12 months” feature for his credit card? 

5. Suppose the annual fees for attending a public university were $7000 in 2010 and the annual cost increase is shown in the graph at right.  Note that x represents the number of years after 2010.

a. Write an equation to describe this situation. Then find out the cost of attending the year you plan to attend college. 

b. What was the cost in 2000, assuming rate of increase was the same during the time period from 2000 to 2010? 

c. In 2012 the annual cost was actually $8244.  How accurate was the model?  What actually happened?  

6. Write the equation for the following graph.

7. Find an exponential function that passes through each pair of points.

a. (−1, −2) and (3, −162) b. (2, 1.75) and (−2, 28) c. (0,5) and (3, 320)

�108. Tickets for a concert have been in incredibly high demand, and as the date for the concert draws closer, the

price of tickets increases exponentially.  The cost of a pair of concert tickets was $150 yesterday, and today it is $162.

a. What is the daily percent rate of increase?  What is the multiplier?

b. What will be the cost of a pair of concert tickets one week from now?  

c. What was the cost of a pair of tickets two weeks ago?   

9. Dusty won $125,000 on the Who Wants to be a Zillionaire? game show.  He decides to place the money into an account that earns 6.25% interest compounded annually and plans not to use any of it until he retires.

a. Write an expression that represents how much money Dusty will have in t years.  

b. How much money will be in the account when he retires in 23 years? 

10. Write the equation for the following graph.

11. Kristin’s grandparents started a savings account for her when she was born.  They invested $500 in an account that pays 8% interest compounded annually.

a. Write an equation to model the amount of money in the account on Kristin’s  xth birthday.

b. How much money is in the account on Kristin’s 16th birthday?  

�1112. A major technology company, ExpoGrow, is growing incredibly fast.  The latest report said that so far, the

number of employees, y, could be found with the equation y = 3(4)x , where x represents the number of years since the company was founded. 

a. How many people founded the company? 

b. How can the growth of this company be described?

c. As part of a major scandal, it was discovered that several statements in the report for ExpoGrow in part (b) were false.  If the company actually had five founders and doubles in size each year, what equation should it have printed in its report?   

13. After paying $20,000 for a car, you read that this model has decreased in value 15% per year over the last several years.  If this trend continues, how much will the car be worth 5 years from now?  

14. Assume that a DVD loses 60% of its value every year it is in a video store.  Suppose the initial value of the DVD was $80.

a. Write an equation that you would use to calculate the video’s new values?  

b. What is the value of the DVD after one year?  After four years?  

c. When does the video have no value? 

15. Aaron just inherited $8000 form his grandmother. He plans to invest the money, not touching it for twenty years. He can invest in treasury bills at 6.67% interest, compounded annually, or in a money market account earning 6.5% annual interest, compounded weekly. Which should he choose? Justify.

�1216. An account is earning 5% annually interest, compounded quarterly. How much will it be worth in 8 years, if

an initial investment of $5000 is put into the account and no withdrawals are made?

a. Write an equation to find the value of the account � after any number of quarters x.

b. Find the value of the account in 8 years.

17. Find a possible exponential function of the form � for each of the following conditions.

a. Has a y-intercept (0, 2) and a multiplier of 0.8.

b. Passes through the points (0, 3.5) and (2, 31.5).

c. Passes through the points (1, 21) and (2, 147).

d. Passes through the points (2, 64) and (-1, 125).

e. Passes through the points (2, 16) and (6, 256).

f x( )

y = abx

�13

18. For � , evaluate each of the following.

a. � b. � c. � d. �

19. Find the annual interest rate on an account that was worth $200 in 1975 and $415.79 in 1990.

20. Eight years ago, Rudy thought he was making a sound investment by buying $1000 worth of Pro Sports Management stock. Unfortunately, his investment depreciated steadily, losing 15% of its value each year. How much is the stock worth now?

21. Most homes appreciate in value, at varying rates, depending on the home’s location, size, and other factors. But, if the home is used as a rental, the IRS allows the owner to assume that it will depreciate in value. Suppose a house that costs $150,000 is used as a rental property, and depreciates as a rate of 8% per year.

a. What is the equation that will give the value of the house?

b. What is the value after 10 years?

c. When will the house be worth half of its purchase price?

22. Casey bought a car for $12,000 ten years ago, and it has been depreciating at a rate of 8% per year since she drove it off the showroom floor.

a. Write an equation to model the amount of money the car is worth at time t.

b. How much is the car worth today?

f x( ) = 20 12

⎛⎝⎜

⎞⎠⎟x

f −2( ) f 0( ) f 2( ) f x( ) = 58

�1423. The number of fans in Lambeau Field after a Packer Game decreases at a rate of 10% per minute after

the end of the game. There were 69,782 fans attending last Sunday’s game.

a. Write a general equation.

b. How many fans were left in the stadium 5 minutes after the game ended?

c. 8 minutes?

Exponential Equations with a Twist

- Every exponential problem you have worked to this point has had a horizontal asymptote of y = 0, aka the _______________ . Because of where the asymptote was located it never effected our exponential equation. There is a formula that is used for situations like this. It is � , where c is your horizontal _______________ .

24. Fall came early in Piney Orchard, and the community swimming pool was still full when the first frost froze the leaves. The outside temperature hovered at 30°. Maintenance quickly turned off the heat so that energy would not be wasted heating a pool that nobody would be swimming in for at least six months. As Tess walked by the pool each day on her way to school, she would peer through the fence at the slowly cooling pool. She could just make out the thermometer across the deck that displayed the water’s temperature. On the first day, she noted that the water temperature was 68°. Four days later, the temperature reading was 58°.

a. Write an equation that models this data.

b. If the outside temperature remains at 30°, and the pool is allowed to cool, how long before it freezes?

25. THE CASE OF THE COOLING CORPSE

The coroner's office is kept at a cool 17°C. Agent 008 kept pacing back and forth trying to keep warm as he waited for any new information about his latest case. For more than three hours now, Dr. Dedman had been performing an autopsy on the Sideroad Slasher's latest victim, and Agent 008 could see that the temperature of the room and the deafening silence were beginning to irritate even Dr. Dedman. The Slasher had been creating more work than Dr. Dedman cared to investigate.

“Dr. Dedman, don't you need to take a break?” Agent 008 queried. “ You've been examining this body for hours! Even if there were any clues, you probably wouldn't see them at this point.”

y = abx + c

�15“I don't know,” Dr. Dedman replied, “ I just have this feeling something is not quite right. Somehow the Slasher slipped up with this one and left a clue. We just have to find it.”

“Well, I have to check in with headquarters,” 008 stated. “ Do you mind if I step out for a couple of hours?”

“No, that's fine,” Dr. Dedman responded. “ Maybe I'll have something by the time you return.”

“Sure,” 008 thought to himself. “ Someone always wants to be the hero and solve everything himself. The doctor just does not realize how big this case really is. The Slasher has left a trail of dead bodies through five states!” Agent 008 left, closing the door quietly. As he walked down the hall, he could hear the doctor's voice describing the victim's gruesome appearance into the tape recorder fade away.

The hallway from the coroner's office to the elevator was long and dark. This was the only way to Dr. Dedman's office. Didn't this frighten most people? Well, it didn't seem to bother old Ajax Boraxo who was busy mopping the floor, thought 008.

He stopped briefly to use the restroom and bumped into one of the deputy coroners, who asked, “ Dedman still at it?”

“Sure is, Dr. Quincy. He's totally obsessed. He's certain there is a clue.” As usual, when leaving the courthouse, 008 had to sign out.

“How's it going down there, Agent 008?” Sergeant Foust asked. Foust spent most of his shifts monitoring the front door, forcing all visitors to sign in while he recorded the time next to the signature. Agent 008 wondered if Foust longed for a more exciting aspect of law enforcement. He thought if he were doing Foust's job he would get a little stir-crazy sitting behind a desk most of the day. Why would someone become a cop to do this?

“Dr. Dedman is convinced he will find something soon. We'll see!” Agent 008 responded. He noticed the time: ten minutes before 2:00. Would he make it to headquarters before the chief left?

“Well, good luck!” Foust shouted as 008 headed out the door.

Some time later, Agent 008 sighed deeply as he returned to the coroner's office. Foust gave his usual greeting: “ Would the secret guest please sign in?” he would say, handing a pen to 008 as he walked through the door. “ Sign in again,” he thought to himself. “Annoying!” 5:05 PM. Agent 008 had not planned to be gone so long, but he had been caught up in what the staff at headquarters had discovered about that calculator he had found. For a moment he saw a positive point to having anyone who came in or out of the courthouse sign in: He knew by quickly scanning the list that Dr. Dedman had not left. In fact, the old guy must still be working on the case.

As he approached the coroner's office, he had a strange feeling that something was wrong. He could not hear or see Dr. Dedman. When he opened the door, the sight inside stopped him in his tracks. Evidently, Dr. Dedman was now the newest victim of the Slasher. But wait! The other body, the one the doctor had been working on, was gone! Immediately, the security desk with its annoying sign-in sheet came to mind. Yes, there were lots of names on that list, but if he could determine the time of Dr. Dedman's death, he might be able to scan the roster to find the murderer! Quickly, he grabbed the thermometer to measure the Doctor's body temperature. He turned around and hit the security buzzer. The bells were deafening. He knew the building would be sealed off instantly and security would be there within seconds.

“Oh no!” Foust cried as he rushed in, “ How could this happen? I spoke to the Doctor less than an hour ago!”

As the security officers crowded into the room, Agent 008 explained what he knew, which was almost nothing. He had stopped long enough to check the doctor's body temperature: 27°C. That was 10°C below normal. Then he remembered: the tape recorder! Dr. Dedman had been taping his observations; that was standard procedure. They began looking everywhere.

�16The Slasher must have realized that the doctor had been taping and taken the tape recorder as well. Exactly an hour had passed during the search, and Agent 008 noticed that the thermometer still remained in Dr. Dedman's side. The thermometer clearly read 24°C. Agent 008 knew he could now determine the time of death.

- What is the temperature of the coroner’s office?

- What time did Agent 008 sign out?

- What time did he come back?

- What is the normal body temp?

- What was Dedman’s body temp the first time it was measured?

- The second time?

a. Find the equation, in the form � , that represents the temperature of the body at any certain time.

b. At approximately what time did Dr. Dedman die?

c. Who is the murderer?

y = abx + c