12.6E: Exercises
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- Aug 24, 2020
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Practice Makes Perfect
Exercise 12.6E.17 Solve Logarithmic Equations Using the Properties of Logarithms
In the following exercises, solve for x.
- log464=2log4x
- log49=2logx
- 3log3x=log327
- 3log6x=log664
- log5(4x−2)=log510
- log3(x2+3)=log34x
- log3x+log3x=2
- log4x+log4x=3
- log2x+log2(x−3)=2
- log3x+log3(x+6)=3
- logx+log(x+3)=1
- logx+log(x−15)=2
- log(x+4)−log(5x+12)=−logx
- log(x−1)−log(x+3)=log1x
- log5(x+3)+log5(x−6)=log510
- log5(x+1)+log5(x−5)=log57
- log3(2x−1)=log3(x+3)+log33
- log(5x+1)=log(x+3)+log2
- Answer
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2. x=7
4. x=4
6. x=1,x=3
8. x=8
10. x=3
12. x=20
14. x=3
16. x=6
18. x=53
Exercise 12.6E.18 Solve Exponential Equations Using Logarithms
In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.
- 3x=89
- 2x=74
- 5x=110
- 4x=112
- ex=16
- ex=8
- (12)x=6
- (13)x=8
- 4ex+1=16
- 3ex+2=9
- 6e2x=24
- 2e3x=32
- 14ex=3
- 13ex=2
- ex+1+2=16
- ex−1+4=12
- Answer
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2. x=log74log2≈6.209
4. x=log112log4≈3.404
6. x=ln8≈2.079
8. x=log8log13≈−1.893
10. x=ln3−2≈−0.901
12. x=ln163≈0.924
14. x=ln6≈1.792
16. x=ln8+1≈3.079
Exercise 12.6E.19 Solve Exponential Equations Using Logarithms
In the following exercises, solve each equation.
- 33x+1=81
- 64x−17=216
- ex2e14=e5x
- ex2ex=e20
- loga64=2
- loga81=4
- lnx=−8
- lnx=9
- log5(3x−8)=2
- log4(7x+15)=3
- lne5x=30
- lne6x=18
- 3logx=log125
- 7log3x=log3128
- log6x+log6(x−5)=log624
- log9x+log9(x−4)=log912
- log2(x+2)−log2(2x+9)=−log2x
- log6(x+1)−log6(4x+10)=log61x
- Answer
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2. x=5
4. x=−4,x=5
6. a=3
8. x=e9
10. x=7
12. x=3
14. x=2
16. x=6
18. x=5
Exercise 12.6E.20 Solve Exponential Equations Using Logarithms
In the following exercises, solve for x, giving an exact answer as well as an approximation to three decimal places.
- 6x=91
- (12)x=10
- 7ex−3=35
- 8ex+5=56
- Answer
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2. x=log10log12≈−3.322
4. x=ln7−5≈−3.054
Exercise 12.6E.21 Use Exponential Models in Applications
In the following exercises, solve.
- Sung Lee invests $5,000 at age 18. He hopes the investments will be worth $10,000 when he turns 25. If the interest compounds continuously, approximately what rate of growth will he need to achieve his goal? Is that a reasonable expectation?
- Alice invests $15,000 at age 30 from the signing bonus of her new job. She hopes the investments will be worth $30,000 when she turns 40. If the interest compounds continuously, approximately what rate of growth will she need to achieve her goal?
- Coralee invests $5,000 in an account that compounds interest monthly and earns 7%. How long will it take for her money to double?
- Simone invests $8,000 in an account that compounds interest quarterly and earns 5%. How long will it take for his money to double?
- Researchers recorded that a certain bacteria population declined from 100,000 to 100 in 24 hours. At this rate of decay, how many bacteria will there be in 16 hours?
- Researchers recorded that a certain bacteria population declined from 800,000 to 500,000 in 6 hours after the administration of medication. At this rate of decay, how many bacteria will there be in 24 hours?
- A virus takes 6 days to double its original population (A=2A0). How long will it take to triple its population?
- A bacteria doubles its original population in 24 hours (A=2A0). How big will its population be in 72 hours?
- Carbon-14 is used for archeological carbon dating. Its half-life is 5,730 years. How much of a 100-gram sample of Carbon-14 will be left in 1000 years?
- Radioactive technetium-99m is often used in diagnostic medicine as it has a relatively short half-life but lasts long enough to get the needed testing done on the patient. If its half-life is 6 hours, how much of the radioactive material form a 0.5 ml injection will be in the body in 24 hours?
- Answer
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2. 6.9%
4. 13.9 years
6. 122,070 bacteria
8. 8 times as large as the original population
10. 0.03 mL
Exercise 12.6E.22 Writing Exercises
- Explain the method you would use to solve these equations: 3x+1=81, 3x+1=75. Does your method require logarithms for both equations? Why or why not?
- What is the difference between the equation for exponential growth versus the equation for exponential decay?
- Answer
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2. Answers will vary.
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

b. After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?