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Logarithms and Logarithmic Functions

Write each equation in logarithmic form.

1. 23 = 8 1092 8 = 3

2. 32 = 9 1093 9 = 2

3.8-2 = 614 1098 6~ = -2

i (tY t 4. = 109~ == 2

Write each equation in exponential form.

5. logs 243 = 5 35 = 243

6. log; 64 = 3 43 = 64

1

7. log93 = "12 9'- = 3

1 __ 5-2 =...!..

8. log5 25 - 2

25

Evaluate each expression. 9. log525 2

10.log93 21"

?.....?

11.loglo 1000 3

12. log1255 31"

13.log4 614 -3

14.log5 6215 -4

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15. logs 83 3

1 _.1.

16.log27"3 3

I

Solve each equation or inequality. Check your solutions.

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17. logsx = 5 243

18.log2x = 38

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19.log. y < 0 0 < y < 1

1 20. logp = 3 64

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21. log2n > -2 n > ~1

t 22. 10gb3 = 9

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23.log6 (4x + 12) = 2 6

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25. logs (x + 2) = logs (3x)

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Chapter 9

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? 24.log2 (4x - 4) > 5 x 9

26. log6(3y - 5) '" log6(2y + 3) y ~ 8

16

Glencoe Algebra 2

NAMI

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Practice

Logarithms and Logarithmic Function

Write each equation in logarithmic form.

1.53 = 125 1095125 = 3

4.3-4

1 = 81

1093811---4

2.70=110971 =0

5.

(4I)"S

=

1

64

I09 .1? 614 -- 3

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Write each equation in exponential form.

7.log6 216 = 3 63 = 216

8.log264=626=64

10.loglO0.00001 = -5 10-5 = 0.00001

11. log255 = "12

1

252 = 5

I 0. IOfJa HI

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12. ~ IOf(,,~ Ii

328 ?? 8

Evaluate each expression.

13.logs 81 4

17.log91 0

t 21. logs -1

14. loglO0.0001 -4

18.logs 4 32"

22.log4 2516 -4

15. log2 116 -4

141. IOll,i '

I9 19.1og7 -2

20. IOHUIl'I .,

+ 23. logg9(" + 1) n 1 24. 2111~.n~ 3

Solve each equation or inequality. Check your solutions.

1 25.loglOn = -3 1000

26. log4x> 3 x > 64

27. log. x

38

2

28.logp = -3 125

29. log, q < 0 0 < q < 1 30. log6(2y I 8), ~ Y

31. logy16 = -4 21"

i 32. log" = -3 2

33. 10gb1024 5 4

34. logs(3x + 7) < logs(7x + 4) 35. log, (8x + 20) = log, (x + "6) 36. logs(x2 - 2) i(1f{1I \

x>~ 3

~

2

37. SOUND An equation for loudness, in decibels,is L = 10 loglOR, where R is Lhl1 ",,1111 IV" intensity of the sound. Sounds that reach levels of 120 decibels or more are pui1I1\t! III humans. What is the relative intensity of 120 decibels? 1012

38. INVESTING Maria invests $1000 in a savings account that pays 4% interest compoundedannually. The value of the accountA at the end offive years can bo determined from the equation logA = log[1000(1+ 0.04)5],Find the value orA Ll1 I.ltl' nearest dollar. $1217

Chapter 9

17

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Answers

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DATE

PERIOD

Practice

Properties of Logarithms

= Use loglO 5 0.6990

expression.

1. 10glO 35 1.5441

= and loglO 7 0.8451

2.loglO 25 1.3980

to approximate the

3.loglO ~ 0.1461

value of each

% 4. 10gi0 -0.1461

5.loglO 245 2.3892 6.loglO 175 2.2431 7.loglO .0.2 -0.6990 8.loglO 275 0.5529

Solve each equation. Check your solutions.

i 9. log, n = log, 8 4

t 10. 10giO u = 10giO 4 8

11. log6 X + log6 9 = log6 54 6

12. logs 48 - logs w = logs 4 12

13. logg (3u + 14) - logg 5 = logg 2u 2 15. log3 y = -log3 16 + '31 log3 64 41'

14.4 log2 x + log2 5 = log2 405 3 16. log2 d = 5 log2 2 - log2 8 4

17.loglO (3m - 5) + 10glO m = 10giO2 2

18.loglO (b + 3) + 10giO b = 10glO4

19. logs (t + 10) - logs (t - 1) = logs 12 2 20. log3 (a + 3) + log3 (a + 2) = log3 6 0

21. 10glO (r + 4) - 10giOr = 10giO (r + 1) 2 22. log, (x2 - 4) - log, (x + 2) = log4 1 3

23. 10giO 4 + 10giO w = 2 25 25.3 logs (x2 + 9) - 6 = 0 ?4

24. logs (n - 3) + logs (n + 4) = 1 4

t 26. logl6 (9x + 5) - logl6 (x2 - 1) = 3

27.log6 (2x - 5) + 1 = log6 (7x + 10) 8 29. 10giO (c2 - 1) - 2 = 10giO (c + 1) 101

28. log2 (5y + 2) - 1 = log2 (1 - 2y) 0 30. log, x'+ 2 log, x - log, 3 = log, 72 6

31. SOUND Recall that the loudness L of a sound in decibels is given by L = 10 10giOR,

where R is the sound's relative intensity. If the intensity of a certain sound is tripled, by

how many decibels does the sound increase? about 4.8 db

.

32. EARTHQUAKES An earthquake rated at 3.5 on the Richter scale is felt by many people,

and an earthquake rated at '4.5 may cause local damage. The Richter scale magnitude reading m is given by m = 10glOx, where x represents the amplitude of the seismic wave causing ground motion. How many times greater is the amplitude of an earthquake that

measures 4.5 on the Richter scale than one that measures 3.5? 10 times

Chapter 9

24

Glencoe Algebra 2

_________________

DATE

PERIOD

Word Problem Practice

Properties of Logarithms

1. MENTAL COMPUTATION

Jessica has

I SIZE For Exercises 5?7, use the

memorized logs 2 = 0.4307 and logs 3 = following information.

0.6826. Using this information, to the

nearest thousandth, what power of 5 is

equal to 6?

1.113

Alicia wanted to try to quantify the terms puny, tiny, small, medium, large, big, huge, and humongous. She picked a number of objects and classified them with these

adjectives of size. She noticed that the scale

2. POWERS A chemist is formulating an acid. The pH of a solution is given by

-loglO C,

seemed exponential. Therefore, she came up with the following definition. Define S to be

ilog3 V, where V is volume in cubic feet.

where C is the concentration of hydrogen ions. Ifthe concentration of

Then use the following table to find the appropriate adjective.

hydrogen ions is increased by a factor of 100, what happens to the pH of the

5 satisfies

Adjective

solution?The pH decreases by 2.

-2:s5 38 {zlz> 3.6555}

18. 5w + 3 = 17 -1'.2396

19. 3OX' = 50 ?1.0725

20. 5x2 - 3 = 72 ?2.3785

21. 42x = 9" + 1 3.8188

22. 2n + 1 = 52n - 10.9117

Express each logarithm in terms of common logarithms.

value to four decimal places.

23.log512 1091012; 1.5440 24.log8 32 1091032; 1.6667

109105

109108

26. log 18 1091018. 4.1699 27. log '6 109106. 0.8155

2 109102 '

9 109109'

Then approximate its

.

25.log11

9

109109 ; 1091011

0.9163

28. log V8 109108? 0.5343 7 2109107'

29. HORTICULTURE Siberian irises flourish when the concentration of hydrogen ions [H+] in the soil is notless than 1.58 X 10-8 mole per liter. What is the pH of the soil in which these irises will flourish? 7.8 or less

30. ACIDITY The pH of vinegar is 2.9 and the pH of milk is 6.6. How many times greater is the hydrogen ion concentration of vinegar than of milk? about 5000

31. BIOLOGY There are initially 1000 bacteria in a culture. The number of bacteria doubles

each hour. The number of bacteria N present after t hours is N = 1000(2)'. How long will it take the culture to increase to 50,000 bacteria? about 5.6 h

32. SOUND An equation for loudness L in decibels is given by L = 10 log R, where R is the sound's relativeintensity, An air-raid siren can reach 150 decibels and jet engine noise can reach 120 decibels. How many times greater is the relative intensity of the air-raid siren than that of the jet engine noise? 1000

Chapter 9

31

Glencoe Algebra 2

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