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© 2001 McGraw-Hill Companies Properties of Logarithms 11.4 871 11.4 OBJECTIVES 1. Apply the properties of logarithms 2. Evaluate logarithmic expressions with any base 3. Solve applications involving logarithms 4. Estimate the value of an antilogarithm As we mentioned earlier, logarithms were developed as aids to numerical computations. The early utility of the logarithm was due to the properties that we will discuss in this section. Even with the advent of the scientific calculator, that utility remains important today. We can apply these same properties to applications in a variety of areas that lead to exponential or logarithmic equations. Because a logarithm is, by definition, an exponent, it seems reasonable that our knowl- edge of the properties of exponents should lead to useful properties for logarithms. That is, in fact, the case. We start with two basic facts that follow immediately from the definition of the logarithm. NOTE The properties follow from the facts that b 1 b and b 0 1 NOTE For Property 3, f 1 (f (x)) f 1 (b x ) log b b x But in general, for any one-to-one function f, f 1 (f (x)) x NOTE The inverse has “undone” whatever f did to x. For b 0 and b 1, Property 1. log b b 1 Property 2. log b 1 0 Rules and Properties: Properties 1 and 2 of Logarithms Property 3. log b b x x Property 4. b log b x x for x 0 Rules and Properties: Properties 3 and 4 of Logarithms We know that the logarithmic function y log b x and the exponential function y b x are inverses of each other. So, for f (x) b x , we have f 1 (x) log b x. It is important to note that for any one-to-one function f, f 1 ( f (x)) x for any x in domain of f and f ( f 1 (x)) x for any x in domain of f 1 Because f (x) b x is a one-to-one function, we can apply the above to the case in which f (x) b x and f 1 (x) log b x to derive the following. Because logarithms are exponents, we can again turn to the familiar exponent rules to derive some further properties of logarithms. Consider the following. We know that log b M x if and only if M b x and log b N y if and only if N b y
Transcript
Page 1: 11.4 Properties of Logarithms - McGraw Hill Education · 103 1000 log 1000 3 102 100 log 100 2 101 10 log 10 1 100 1log1 0 10 1 0.1 log 0.1 1 10 2 0.01 log 0.01 2 10 3 0.001 log 0.001

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Properties of Logarithms11.4

871

11.4 OBJECTIVES

1. Apply the properties of logarithms2. Evaluate logarithmic expressions with any base3. Solve applications involving logarithms4. Estimate the value of an antilogarithm

As we mentioned earlier, logarithms were developed as aids to numerical computations.The early utility of the logarithm was due to the properties that we will discuss in thissection. Even with the advent of the scientific calculator, that utility remains importanttoday. We can apply these same properties to applications in a variety of areas that leadto exponential or logarithmic equations.

Because a logarithm is, by definition, an exponent, it seems reasonable that our knowl-edge of the properties of exponents should lead to useful properties for logarithms. That is,in fact, the case.

We start with two basic facts that follow immediately from the definition of the logarithm.

NOTE The properties followfrom the facts that

b1 � b and b0 � 1

NOTE For Property 3,

f�1(f(x)) � f�1(bx) � logb bx

But in general, for any one-to-one function f,

f �1(f(x)) � x

NOTE The inverse has“undone” whatever f did to x.

For b � 0 and b � 1,

Property 1. logb b � 1

Property 2. logb 1 � 0

Rules and Properties: Properties 1 and 2 of Logarithms

Property 3. logb bx � x

Property 4. b logb x � x for x � 0

Rules and Properties: Properties 3 and 4 of Logarithms

We know that the logarithmic function y � logb x and the exponential function y � bx areinverses of each other. So, for f(x) � bx, we have f �1(x) � logb x.

It is important to note that for any one-to-one function f,

f �1( f (x)) � x for any x in domain of f

and

f( f �1(x)) � x for any x in domain of f �1

Because f(x) � bx is a one-to-one function, we can apply the above to the case in which

f(x) � bx and f �1(x) � logb x

to derive the following.

Because logarithms are exponents, we can again turn to the familiar exponent rules toderive some further properties of logarithms. Consider the following.

We know that

logb M � x if and only if M � bx

and

logb N � y if and only if N � b y

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Using the Properties of Logarithms

Expand, using the properties of logarithms.

(a) logb xy � logb x � logb y Product property

(b) logb � logb xy � logb z Quotient property

� logb x � logb y � logb z Product property

(c) log10 x2y3 � log10 x2 � log10 y3 Product property

� 2 log10 x � 3 log10 y Power property

(d) logb � logb Definition of rational exponent

� logb Power property

� (logb x � logb y) Quotient property1

2

x

y

1

2

�x

y�1�2Bx

y

xy

z

NOTE In all cases, M, N � 0,b � 0, b � 1, and p is any realnumber.

Example 1

Then

M � N � bx � by � bx�y (1)

From equation (1) we see that x � y is the power to which we must raise b to get the productMN. In logarithmic form, that becomes

logb MN � x � y (2)

Now, because x � logb M and y � logb N, we can substitute in (2) to write

logb MN � logb M � logb N (3)

This is the first of the basic logarithmic properties presented here. The remainingproperties may all be proved by arguments similar to those presented in equations (1) to (3).

Product property

logb MN � logb M � logb N

Quotient property

logb � logb M � logb N

Power property

logb Mp � p logb M

MN

Rules and Properties: Properties of Logarithms

Many applications of logarithms require using these properties to write a single loga-rithmic expression as the sum or difference of simpler expressions, as Example 1 illustrates.

NOTE Recall 1a � a1�2.

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C H E C K Y O U R S E L F 1

Expand each expression, using the properties of logarithms.

(a) logb x2y3z (b) log10Bxy

z

In some cases, we will reverse the process and use the properties to write a single loga-rithm, given a sum or difference of logarithmic expressions.

Example 2

Rewriting Logarithmic Expressions

Write each expression as a single logarithm with coefficient 1.

(a) 2 logb x � 3 logb y

� logb x2 � logb y3 Power property

� logb x2y 3 Product property

(b) (log2 x � log2 y)

� Quotient property

� log2 Power property

� log2 A x

y

�x

y�1�2

1

2 �log2 x

y�

1

2

C H E C K Y O U R S E L F 2

Write each expression as a single logarithm with coefficient 1.

(a) 3 logb x � 2 logb y � 2 logb z (b) (2 log2 x � log2 y)1

3

Example 3 illustrates the basic concept of the use of logarithms as a computational aid.

NOTE We have written thelogarithms correct to threedecimal places and will followthis practice throughout theremainder of this chapter.Keepin mind, however, that this is anapproximation and that 100.301

will only approximate 2. Verifythis with your calculator.

Example 3

Evaluating Logarithmic Expressions

Suppose log10 2 � 0.301 and log10 3 � 0.447. Given these values, find the following.

(a) log10 6 Because 6 � 2 � 3

� log10 (2 � 3)

� log10 2 � log10 3

� 0.301 � 0.477

� 0.778

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NOTE Notice that logb 1 � 0for any base b.

NOTE We have extended theproduct rule for logarithms.

C H E C K Y O U R S E L F 3

Given the values above for log10 2 and log10 3, find each of the following.

(a) log10 12 (b) log10 27 (c) log10 13 2

There are two types of logarithms used most frequently in mathematics:

Logarithms to base 10

Logarithms to base e

Of course, the use of logarithms to base 10 is convenient because our number system hasbase 10. We call logarithms to base 10 common logarithms, and it is customary to omit thebase in writing a common (or base-10) logarithm. So

log N means log10 N

The following table shows the common logarithms for various powers of 10.

Exponential Form Logarithmic Form

103 � 1000 log 1000 � 3102 � 100 log 100 � 2101 � 10 log 10 � 1100 � 1 log 1 � 010�1 � 0.1 log 0.1 � �110�2 � 0.01 log 0.01 � �210�3 � 0.001 log 0.001 � �3

NOTE Verify each answer withyour calculator.

NOTE When no base for “log”is written, it is assumed to be 10.

(b) log10 18 Because 18 � 2 � 3 � 3

� log10 (2 � 3 � 3)

� log10 2 � log10 3 � log10 3

� 1.255

(c) log10 Because 5

� log10

� log10 1 � log10 32

� 0 � 2 log10 3

� �0.954

(d) log10 16 Because 16 � 24

� log10 24 � 4 log10 2

� 1.204

(e) log10 Because �

� log10 � log10 3

� 0.239

1

231�2

31�21313

1

32

132

19

1

9

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Example 4

Approximating Logarithms with a Calculator

Verify each of the following with a calculator.

(a) log 4.8 � 0.681

(b) log 48 � 1.681

(c) log 480 � 2.681

(d) log 4800 � 3.681

(e) log 0.48 � �0.319

NOTE The number 4.8 liesbetween 1 and 10, so log 4.8lies between 0 and 1.

NOTE Notice that

480 � 4.8 � 102

and

log (4.8 � 102)

� log 4.8 � log 102

� log 4.8 � 2

� 2 � log 4.8C H E C K Y O U R S E L F 4

Use your calculator to find each of the following logarithms, correct to three decimalplaces.

(a) log 2.3 (b) log 23 (c) log 230

(d) 2300 (e) log 0.23 (f) log 0.023

Let’s look at an application of common logarithms from chemistry. Common logarithmsare used to define the pH of a solution. This is a scale that measures whether the solution isacidic or basic.

NOTE The value of log 0.48 isreally �1 � 0.681. Yourcalculator will combine thesigned numbers.

NOTE A solution is neutralwith pH � 7, acidic if the pH isless than 7, and basic if the pHis greater than 7.

NOTE Notice the use of theproduct rule here.

NOTE Also, in general,logb bx � x, so log 10�7 � �7.

Example 5

A pH Application

Find the pH of each of the following. Determine whether each is a base or an acid.

(a) Rainwater: [H�] � 1.6 � 10�7

From the definition,

pH � �log [H�]

� �log (1.6 � 10�7)

� �(log 1.6 � log 10�7)

� �[0.204 � (�7)]

� �(�6.796) � 6.796

The rain is just slightly acidic.

The pH of a solution is defined as

pH � �log [H�]

in which [H�] is the hydrogen ion concentration, in moles per liter (mol/L), in the solution.

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NOTE Because it is a one-to-one function, the logarithmicfunction has an inverse.

C H E C K Y O U R S E L F 5

Find the pH for the following solutions. Are they acidic or basic?

(a) Orange juice: [H�] � 6.8 � 10�5

(b) Drain cleaner: [H�] � 5.2 � 10�13

Many applications require reversing the process. That is, given the logarithm of a num-ber, we must be able to find that number. The process is straightforward.

Example 6

Using a Calculator to Estimate Antilogarithms

Suppose that log x � 2.1567. We want to find a number x whose logarithm is 2.1567. Usinga calculator requires one of the following sequences:

2.1567 or 2.1567 or 2.1567

Both give the result 143.45, often called the antilogarithm of 2.1567.

log2ndlogINV10x

C H E C K Y O U R S E L F 6

Find the value of the antilogarithm of x.

(a) log x � 0.828 (b) log x � 1.828

(c) log x � 2.828 (d) log x � �0.172

Let’s return to the application from chemistry for an example requiring the use of theantilogarithm.

(b) Household ammonia: [H�] � 2.3 � 10�8

pH � �log (2.3 � 10�8)

� �(log 2.3 � log 10�8)

� �[0.362 � (�8)]

� 7.638

The ammonia is slightly basic.

(c) Vinegar: [H�] � 2.9 � 10�3

pH � �log (2.9 � 10�3)

� �(log 2.9 � log 10�3)

� 2.538

The vinegar is very acidic.

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Example 7

A pH Application

Suppose that the pH for tomato juice is 6.2. Find the hydrogen ion concentration [H�].Recall from our earlier formula that

pH � �log [H�]

In this case, we have

6.2 � �log [H�] or log [H�] � �6.2

The desired value for [H�] is then the antilogarithm of �6.2.

The result is 0.00000063, and we can write

[H�] � 6.3 � 10�7

C H E C K Y O U R S E L F 7

The pH for eggs is 7.8. Find [H�] for eggs.

As we mentioned, there are two systems of logarithms in common use. The second typeof logarithm uses the number e as a base, and we call logarithms to base e the naturallogarithms. As with common logarithms, a convenient notation has developed, as the fol-lowing definition shows.

NOTE Natural logarithms arealso called napierian logarithmsafter Napier. The importance ofthis system of logarithms wasnot fully understood until laterdevelopments in the calculus.

NOTE The restrictions on thedomain of the naturallogarithmic function are thesame as before. The function isdefined only if x � 0.

The natural logarithm is a logarithm to base e, and it is denoted ln x, as

ln x � loge x

Definitions: Natural Logarithm

By the general definition of a logarithm,

y � ln x means the same as x � ey

and this leads us directly to the following.

ln 1 � 0 because e0 � 1

ln e � 1 because e1 � e

ln e2 � 2 and ln e�3 � �3

Example 8

Estimating Natural Logarithms

To find other natural logarithms, we can again turn to a calculator. To find the value of ln 2,use the sequence

2 or 2

The result is 0.693 (to three decimal places).

)ln (ln

NOTE In general

logb bx � x b � 1

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C H E C K Y O U R S E L F 8

Use a calculator to find each of the following.

(a) ln 3 (b) ln 6 (c) ln 4 (d) ln 13

Of course, the properties of logarithms are applied in an identical fashion, no matterwhat the base.

Example 9

Evaluating Logarithms

If ln 2 � 0.693 and ln 3 � 1.099, find the following.

(a) ln 6 � ln (2 � 3) � ln 2 � ln 3 � 1.792

(b) ln 4 � ln 22 � 2 ln 2 � 1.386

(c) ln � ln ln 3 � 0.549

Again, verify these results with your calculator.

31�2 �1

213

C H E C K Y O U R S E L F 9

Use In 2 � 0.693 and ln 3 � 1.099 to find the following.

(a) ln 12 (b) ln 27

The natural logarithm function plays an important role in both theoretical and appliedmathematics. Example 10 illustrates just one of the many applications of this function.

NOTE Recall that logb MN � logb M � logb Nlogb Mp � p logb M

NOTE Recall that we read S(t)as “S of t”, which means that Sis a function of t.

Example 10

A Learning Curve Application

A class of students took a final mathematics examination and received an average scoreof 76. In a psychological experiment, the students are retested at weekly intervals overthe same material. If t is measured in weeks, then the new average score after t weeks isgiven by

S(t) � 76 � 5 ln (t � 1)

Complete the following.

(a) Find the score after 10 weeks.

S(t) � 76 � 5 ln (10 � 1)

� 76 � 5 ln 11 � 64

S

t

80

60

40

20

10 20 30

NOTE This is an example of aforgetting curve. Note how itdrops more rapidly at first.Compare this curve to thelearning curve drawn inSection 11.2, exercise 62.

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C H E C K Y O U R S E L F 1 0

The average score for a group of biology students, retested after time t (in months),is given by

S(t) � 83 � 9 ln (t � 1)

Find the average score after

(a) 3 months (b) 6 months

We conclude this section with one final property of logarithms. This property will allowus to quickly find the logarithm of a number to any base. Although work with logarithmswith bases other than 10 or e is relatively infrequent, the relationship between logarithms ofdifferent bases is interesting in itself. Consider the following argument.

Suppose that

x � log2 5

or

2x � 5 (4)

Taking the logarithm to base 10 of both sides of equation (4) yields

log 2x � log 5

or

x log 2 � log 5 Use the power property of logarithms. (5)

(Note that we omit the 10 for the base and write log 2, for example.) Now, dividing bothsides of equation (5) by log 2, we have

We can now find a value for x with the calculator. Dividing with the calculator log 5 bylog 2, we get an approximate answer of 2.3219.

Because x � log2 5 and x � then

Generalizing our result, we find the following.

log2 5 �log 5

log 2

log 5

log 2,

x �log 5

log 2

(b) Find the score after 20 weeks.

S(t) � 76 � 5 ln (20 � 1) � 61

(c) Find the score after 30 weeks.

S(t) � 76 � 5 ln (30 � 1) � 59

For the positive real numbers a and x,

loga x �log xlog a

Rules and Properties: Change-of-Base Formula

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Example 11

Evaluating Logarithms

Find log5 15.From the change-of-base formula with a � 5 and b � 10,

log5 15 �

� 1.683

The calculator sequence for the above computation is

15 5 ENTERloglog

log 15

log 5NOTE We have written log 15rather than log 15 to emphasizethe change-of-base formula.

NOTE log5 5 � 1 andlog5 25 � 2, so the result forlog5 15 must be between 1and 2.

C H E C K Y O U R S E L F 1 1

Use the change-of-base formula to find log8 32.

Note that the logarithm on the left side has base a whereas the logarithms on the right sidehave base 10. This allows us to calculate the logarithm to base a of any positive number,given the corresponding logarithms to base 10 (or any other base), as Example 11 illustrates.

C A U T I O N

A common error is to write

� log 15 � log 5

This is not a logarithmicproperty. A true statementwould be

log � log 15 � log 5

but

log and

are not the same.

log 15log 5

155

155

log 15log 5

C H E C K Y O U R S E L F A N S W E R S

1. (a) 2 logb x � 3 logb y � logb z; (b) (log10 x � log10 y � log10 z)

2. (a) (b) log2 3. (a) 1.079; (b) 1.431; (c) 0.100

4. (a) 0.362; (b) 1.362; (c) 2.362; (d) 3.362; (e) �0.638; (f ) �1.638

5. (a) 4.17, acidic; (b) 12.28, basic 6. (a) 6.73; (b) 67.3; (c) 673; (d) 0.673

7. [H�] � 1.6 � 10�8 8. (a) 1.099; (b) 1.792; (c) 1.386; (d) 0.549

9. (a) 2.485; (b) 3.297 10. (a) 70.5; (b) 65.5 11. log8 32 �log 32

log 8� 1.667

B3 x2

ylogb

x3y2

z2 ;

1

2

Note: Recall that the loge x is called the natural log of x. We use “ln x” to designate thenatural log of x. A special case of the change-of-base formula allows us to find natural log-arithms in terms of common logarithms:

ln x �

so

ln � 2.304, then ln x � 2.304 log x

Of course, because all modern calculators have both the log function key and the lnfunction key, this conversion formula is now rarely used.

x �log x

0.434 or, because

1

0.434

log x

log e

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Exercises

In exercises 1 to 18, use the properties of logarithms to expand each expression.

1. logb 5x 2. log3 7x

3. log4 4. logb

5. log3 a2 6. log5 y4

7. log5 8. log

9. logb x3y2 10. log5 x2z4

11. log4 y 2 12. logb x3

13. logb 14. log5

15. log 16. log4

17. log5 18. logbB4 x2y

z3A3 xy

z2

x31y

z 2 xy21z

3

xy

x2y

z

13 z1x

13 z1x

2

y

x

3

11.4

Name

Section Date

ANSWERS

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

881

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In exercises 19 to 30, write each expression as a single logarithm.

19. logb x � logb y 20. log5 x � log5 y

21. 2 log2 x � log2 y 22. 3 logb x � logb z

23. logb x � logb y 24. logb x � 2 logb z

25. logb x � 2 logb y � logb z 26. 2 log5 x � (3 log5 y � log5 z)

27. log6 y � 3 log6 z 28. logb x � logb y � 4 logb z

29. (2 logb x � logb y � logb z) 30. (2 log4 x � log4 y � 3 log4 z)

In exercises 31 to 38, given that log 2 � 0.301 and log 3 � 0.477, find each logarithm.

31. log 24 32. log 36

33. log 8 34. log 81

35. log 36. log

37. log 38. log

In exercises 39 to 44, use your calculator to find each logarithm.

39. log 6.8 40. log 68

41. log 680 42. log 6800

43. log 0.68 44. log 0.068

1

27

1

4

13 312

1

5

1

3

1

3

1

2

1

3

1

2

ANSWERS

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

42.

43.

44.

882

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In exercises 45 and 46, find the pH, given the hydrogen ion concentration [H�] for eachsolution. Use the formula

pH � �log [H�]

Are the solutions acidic or basic?

45. Blood: [H�] � 3.8 � 10�8 46. Lemon juice: [H�] � 6.4 � 10�3

In exercises 47 to 50, use your calculator to find the antilogarithm for each logarithm.

47. 0.749 48. 1.749

49. 3.749 50. �0.251

In exercises 51 and 52, given the pH of the solutions, find the hydrogen ion concentration[H�].

51. Wine: pH � 4.7 52. Household ammonia: pH � 7.8

In exercises 53 to 56, use your calculator to find each logarithm.

53. ln 2 54. ln 3

55. ln 10 56. ln 30

The average score on a final examination for a group of psychology students, retestedafter time t (in weeks), is given by

S � 85 � 8 ln (t � 1)

In exercises 57 and 58, find the average score on the retests:

57. After 3 weeks 58. After 12 weeks

In exercises 59 and 60, use the change-of-base formula to find each logarithm.

59. log3 25 60. log5 30© 2

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sANSWERS

45.

46.

47.

48.

49.

50.

51.

52.

53.

54.

55.

56.

57.

58.

59.

60.

883

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The amount of a radioactive substance remaining after a given amount of time t is givenby the following formula:

A � elt�ln A0

in which A is the amount remaining after time t, variable A0 is the original amount of thesubstance, and l is the radioactive decay constant.

61. How much plutonium 239 will remain after 50,000 years if 24 kg was originallystored? Plutonium 239 has a radioactive decay constant of �0.000029.

62. How much plutonium 241 will remain after 100 years if 52 kg was originally stored?Plutonium 241 has a radioactive decay constant of �0.053319.

63. How much strontium 90 was originally stored if after 56 years it is discovered that15 kg still remains? Strontium 90 has a radioactive decay constant of �0.024755.

64. How much cesium 137 was originally stored if after 90 years it is discovered that20 kg still remains? Cesium 137 has a radioactive decay constant of �0.023105.

65. Which keys on your calculator are function keys and which are operation keys? Whatis the difference?

66. How is the pH factor relevant to your selection of a hair care product?

Answers

1. logb 5 � logb x 3. log4 x � log4 3 5. 2 log3 a 7.

9. 3 logb x � 2 logb y 11. 2 log4 y � log4 x 13. 2 logb x � logb y � logb z

15. log x � 2 log y � log z 17. 19. logb xy

21. log2 23. logb x 25. logb 27. log6

29. logb 31. 1.380 33. 0.903 35. 0.151 37. �0.602

39. 0.833 41. 2.833 43. �0.167 45. 7.42, basic 47. 5.6149. 5610 51. 2 � 10�5 53. 0.693 55. 2.303 57. 7459. 2.930 61. 5.6 kg 63. 60 kg 65.

B3 x2y

z

1y

z3

xy2

z1y

x2

y

1

3(log5 x � log5 y � 2 log5 z)

1

2

1

2

1

2 log5 x

ANSWERS

61.

62.

63.

64.

65.

66.

884


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