11.4
Properties of Logarithms
11.4
OBJECTIVES
1.
2.
3.
4.
Apply the properties of logarithms
Evaluate logarithmic expressions with any base
Solve applications involving logarithms
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 knowledge 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.
Rules and Properties:
NOTE The properties follow
from the facts that
b1 b and b0 1
Properties 1 and 2 of Logarithms
For b 0 and b 1,
Property 1. logb b 1
Property 2. logb 1 0
NOTE The inverse has
“undone” whatever f did to x.
We know that the logarithmic function y logb x and the exponential function y bx are
inverses 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.
Rules and Properties:
Property 3. logb b x x
© 2001 McGraw-Hill Companies
NOTE For Property 3,
f
1
(f(x)) f
1
(b ) logb b
x
Properties 3 and 4 of Logarithms
x
Property 4. b logb x x
for x 0
But in general, for any
one-to-one function f,
f 1(f(x)) x
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
logb M x
if and only if
M bx
if and only if
N by
and
logb N y
871
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CHAPTER 11
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Then
M N b x b y b xy
(1)
From equation (1) we see that x y is the power to which we must raise b to get the product
MN. 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 remaining
properties may all be proved by arguments similar to those presented in equations (1) to (3).
Rules and Properties:
NOTE In all cases, M, N 0,
b 0, b 1, and p is any real
number.
Properties of Logarithms
Product property
logb MN logb M logb N
Quotient property
logb
M
logb M logb N
N
Power property
logb M p p logb M
Many applications of logarithms require using these properties to write a single logarithmic expression as the sum or difference of simpler expressions, as Example 1 illustrates.
Example 1
Using the Properties of Logarithms
Expand, using the properties of logarithms.
(b) logb
xy
logb xy logb z
z
logb x logb y logb z
(c) log10 x2y3 log10 x2 log10 y3
2 log10 x 3 log10 y
NOTE Recall 1a a12.
(d) logb
x
x
logb
By
y
Product property
Quotient property
Product property
Product property
Power property
12
Definition of rational exponent
1
x
logb
2
y
Power property
1
(logb x logb y)
2
Quotient property
© 2001 McGraw-Hill Companies
(a) logb xy logb x logb y
PROPERTIES OF LOGARITHMS
SECTION 11.4
873
CHECK YOURSELF 1
Expand each expression, using the properties of logarithms.
(a) logb x2y 3z
(b) log10
xy
Bz
In some cases, we will reverse the process and use the properties to write a single logarithm, 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 x y
Product property
2 3
(b)
1
(log2 x log2 y)
2
1
x
log2
2
y
12
log2
log2
x
Ay
x
y
Quotient property
Power property
CHECK YOURSELF 2
Write each expression as a single logarithm with coefficient 1.
(a) 3 logb x 2 logb y 2 logb z
(b)
1
(2 log2 x log2 y)
3
Example 3 illustrates the basic concept of the use of logarithms as a computational aid.
© 2001 McGraw-Hill Companies
Example 3
Evaluating Logarithmic Expressions
NOTE We have written the
logarithms correct to three
decimal places and will follow
this practice throughout the
remainder of this chapter.Keep
in mind, however, that this is an
approximation and that 100.301
will only approximate 2. Verify
this with your calculator.
Suppose log10 2 0.301 and log10 3 0.447. Given these values, find the following.
(a) log10 6
log10 (2 3)
log10 2 log10 3
0.301 0.477
0.778
Because 6 2 3
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CHAPTER 11
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Because 18 2 3 3
(b) log10 18
log10 (2 3 3)
NOTE We have extended the
log10 2 log10 3 log10 3
product rule for logarithms.
1.255
(c) log10
1
9
log10
NOTE Notice that logb 1 0
for any base b.
Because
1 1
5
9 32
1
32
log10 1 log10 32
0 2 log10 3
0.954
Because 16 24
(d) log10 16
log10 24 4 log10 2
1.204
NOTE Verify each answer with
your calculator.
(e) log10 13
log10 312 Because 13 312
1
log10 3
2
0.239
CHECK YOURSELF 3
Given the values above for log10 2 and log10 3, find each of the following.
(b) log10 27
(a) log10 12
3
(c) log10 12
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 has
base 10. We call logarithms to base 10 common logarithms, and it is customary to omit the
base 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
102 100
101 10
100 1
101 0.1
102 0.01
103 0.001
log 1000
3
log 100
2
log 10
1
log
1
0
log
0.1 1
log
0.01 2
log
0.001 3
© 2001 McGraw-Hill Companies
NOTE When no base for “log”
is written, it is assumed to be 10.
PROPERTIES OF LOGARITHMS
SECTION 11.4
875
Example 4
Approximating Logarithms with a Calculator
Verify each of the following with a calculator.
NOTE The number 4.8 lies
between 1 and 10, so log 4.8
lies between 0 and 1.
NOTE Notice that
480 4.8 102
and
(a)
(b)
(c)
(d)
(e)
log 4.8 0.681
log 48 1.681
log 480 2.681
log 4800 3.681
log 0.48 0.319
log (4.8 102)
log 4.8 log 102
log 4.8 2
2 log 4.8
CHECK YOURSELF 4
Use your calculator to find each of the following logarithms, correct to three decimal
places.
NOTE The value of log 0.48 is
really 1 0.681. Your
calculator will combine the
signed numbers.
(a) log 2.3
(d) 2300
(b) log 23
(e) log 0.23
(c) log 230
(f) log 0.023
Let’s look at an application of common logarithms from chemistry. Common logarithms
are used to define the pH of a solution. This is a scale that measures whether the solution is
acidic or basic.
NOTE A solution is neutral
with pH 7, acidic if the pH is
less than 7, and basic if the pH
is greater than 7.
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.
Example 5
A pH Application
Find the pH of each of the following. Determine whether each is a base or an acid.
© 2001 McGraw-Hill Companies
(a) Rainwater: [H] 1.6 107
From the definition,
pH log [H]
log (1.6 107)
NOTE Notice the use of the
product rule here.
NOTE Also, in general,
logb bx x, so log 107 7.
(log 1.6 log 107)
[0.204 (7)]
(6.796) 6.796
The rain is just slightly acidic.
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CHAPTER 11
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
(b) Household ammonia: [H] 2.3 108
pH log (2.3 108)
(log 2.3 log 108)
[0.362 (8)]
7.638
The ammonia is slightly basic.
(c) Vinegar: [H] 2.9 103
pH log (2.9 103)
(log 2.9 log 103)
2.538
The vinegar is very acidic.
CHECK YOURSELF 5
Find the pH for the following solutions. Are they acidic or basic?
(a) Orange juice: [H] 6.8 105
(b) Drain cleaner: [H] 5.2 1013
Many applications require reversing the process. That is, given the logarithm of a number, 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. Using
a calculator requires one of the following sequences:
NOTE Because it is a one-to-
2.1567 10 x
one function, the logarithmic
function has an inverse.
Both give the result 143.45, often called the antilogarithm of 2.1567.
2.1567 INV log
or
2nd log 2.1567
CHECK YOURSELF 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 the
antilogarithm.
© 2001 McGraw-Hill Companies
or
PROPERTIES OF LOGARITHMS
SECTION 11.4
877
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]
log [H] 6.2
or
The desired value for [H] is then the antilogarithm of 6.2.
The result is 0.00000063, and we can write
[H] 6.3 107
CHECK YOURSELF 7
The pH for eggs is 7.8. Find [H] for eggs.
NOTE Natural logarithms are
also called napierian logarithms
after Napier. The importance of
this system of logarithms was
not fully understood until later
developments in the calculus.
NOTE The restrictions on the
domain of the natural
logarithmic function are the
same as before. The function is
defined only if x 0.
As we mentioned, there are two systems of logarithms in common use. The second type
of logarithm uses the number e as a base, and we call logarithms to base e the natural
logarithms. As with common logarithms, a convenient notation has developed, as the following definition shows.
Definitions: Natural Logarithm
The natural logarithm is a logarithm to base e, and it is denoted ln x, as
ln x loge x
By the general definition of a logarithm,
y ln x
means the same as
x ey
© 2001 McGraw-Hill Companies
and this leads us directly to the following.
ln 1 0
because
e0 1
NOTE In general
ln e 1
because
e1 e
logb bx x
ln e2 2
and
b1
ln e3 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
ln 2
or
ln ( 2 )
The result is 0.693 (to three decimal places).
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CHAPTER 11
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
CHECK YOURSELF 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 matter
what 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
NOTE Recall that
logb MN logb M logb N
logb M p p logb M
(b) ln 4 ln 22 2 ln 2 1.386
(c) ln 13 ln 312 1
ln 3 0.549
2
Again, verify these results with your calculator.
CHECK YOURSELF 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 applied
mathematics. Example 10 illustrates just one of the many applications of this function.
Example 10
NOTE Recall that we read S(t)
as “S of t”, which means that S
is a function of t.
S
80
60
A Learning Curve Application
A class of students took a final mathematics examination and received an average score
of 76. In a psychological experiment, the students are retested at weekly intervals over
the same material. If t is measured in weeks, then the new average score after t weeks is
given by
40
Complete the following.
20
t
10
20
30
NOTE This is an example of a
forgetting curve. Note how it
drops more rapidly at first.
Compare this curve to the
learning curve drawn in
Section 11.2, exercise 62.
(a) Find the score after 10 weeks.
S(t) 76 5 ln (10 1)
76 5 ln 11 64
© 2001 McGraw-Hill Companies
S(t) 76 5 ln (t 1)
PROPERTIES OF LOGARITHMS
SECTION 11.4
879
(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
CHECK YOURSELF 10
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 allow
us to quickly find the logarithm of a number to any base. Although work with logarithms
with bases other than 10 or e is relatively infrequent, the relationship between logarithms of
different 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 both
sides of equation (5) by log 2, we have
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x
log 5
log 2
We can now find a value for x with the calculator. Dividing with the calculator log 5 by
log 2, we get an approximate answer of 2.3219.
log 5
, then
Because x log2 5 and x log 2
log 5
log2 5 log 2
Generalizing our result, we find the following.
Rules and Properties:
Change-of-Base Formula
For the positive real numbers a and x,
loga x log x
log a
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CHAPTER 11
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Note that the logarithm on the left side has base a whereas the logarithms on the right side
have 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.
Example 11
Evaluating Logarithms
Find log5 15.
From the change-of-base formula with a 5 and b 10,
NOTE We have written log 15
log5 15 rather than log 15 to emphasize
the change-of-base formula.
NOTE log5 5 1 and
log5 25 2, so the result for
log5 15 must be between 1
and 2.
log 15
log 5
1.683
The calculator sequence for the above computation is
log 15 log 5 ENTER
CHECK YOURSELF 11
Use the change-of-base formula to find log8 32.
A common error is to write
log 15
log 15 log 5
log 5
This is not a logarithmic
property. A true statement
would be
log
15
log 15 log 5
5
Note: Recall that the loge x is called the natural log of x. We use “ln x” to designate the
natural log of x. A special case of the change-of-base formula allows us to find natural logarithms in terms of common logarithms:
ln x so
ln x but
log
15
5
and
log 15
log 5
are not the same.
log x
log e
1
log x
or, because
2.304, then ln x 2.304 log x
0.434
0.434
Of course, because all modern calculators have both the log function key and the ln
function key, this conversion formula is now rarely used.
CHECK YOURSELF ANSWERS
1. (a) 2 logb x 3 logb y logb z; (b)
2
x3y2
3 x
3. (a) 1.079; (b) 1.431; (c) 0.100
2 ; (b) log2
z
By
(a) 0.362; (b) 1.362; (c) 2.362; (d) 3.362; (e) 0.638; (f ) 1.638
(a) 4.17, acidic; (b) 12.28, basic
6. (a) 6.73; (b) 67.3; (c) 673; (d) 0.673
8
[H ] 1.6 10
8. (a) 1.099; (b) 1.792; (c) 1.386; (d) 0.549
log 32
(a) 2.485; (b) 3.297
10. (a) 70.5; (b) 65.5
11. log8 32 1.667
log 8
2. (a) logb
4.
5.
7.
9.
1
(log10 x log10 y log10 z)
2
© 2001 McGraw-Hill Companies
CAUTION
Name
11.4 Exercises
Section
Date
In exercises 1 to 18, use the properties of logarithms to expand each expression.
1. logb 5x
ANSWERS
2. log3 7x
1.
2.
3. log4
x
3
4. logb
2
y
3.
4.
5.
5. log3 a2
6.
6. log5 y4
7.
8.
7. log5 1x
3
8. log 1z
9.
10.
11.
9. logb x3y2
10. log5 x2z4
12.
13.
11. log4 y 2 1x
3
12. logb x3 1z
14.
15.
13. logb
x2y
z
14. log5
3
xy
16.
© 2001 McGraw-Hill Companies
17.
15. log
xy2
1z
16. log4
x3 1y
z2
17. log5
xy
A z2
18. logb
x2y
B z3
3
18.
4
881
ANSWERS
19.
In exercises 19 to 30, write each expression as a single logarithm.
20.
19. logb x logb y
20. log5 x log5 y
21. 2 log2 x log2 y
22. 3 logb x logb z
21.
22.
23.
23. logb x 24.
1
logb y
2
25. logb x 2 logb y logb z
25.
26.
1
log x 2 logb z
3 b
26. 2 log5 x (3 log5 y log5 z)
27.
1
log6 y 3 log6 z
2
28. logb x 29.
1
(2 logb x logb y logb z)
3
30.
27.
28.
24.
1
logb y 4 logb z
3
1
(2 log4 x log4 y 3 log4 z)
5
29.
In exercises 31 to 38, given that log 2 0.301 and log 3 0.477, find each logarithm.
30.
31. log 24
32. log 36
33. log 8
34. log 81
35. log 12
36. log 13
31.
32.
33.
34.
35.
3
36.
37. log
37.
1
4
38. log
1
27
39.
In exercises 39 to 44, use your calculator to find each logarithm.
40.
39. log 6.8
40. log 68
41. log 680
42. log 6800
43. log 0.68
44. log 0.068
41.
42.
43.
44.
882
© 2001 McGraw-Hill Companies
38.
ANSWERS
In exercises 45 and 46, find the pH, given the hydrogen ion concentration [H] for each
solution. Use the formula
45.
46.
pH log [H]
47.
Are the solutions acidic or basic?
48.
45. Blood: [H] 3.8 108
46. Lemon juice: [H] 6.4 103
49.
50.
In exercises 47 to 50, use your calculator to find the antilogarithm for each logarithm.
47. 0.749
48. 1.749
51.
52.
53.
49. 3.749
50. 0.251
54.
55.
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
56.
57.
58.
59.
In exercises 53 to 56, use your calculator to find each logarithm.
53. ln 2
54. ln 3
55. ln 10
56. ln 30
60.
The average score on a final examination for a group of psychology students, retested
after time t (in weeks), is given by
© 2001 McGraw-Hill Companies
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
883
ANSWERS
61.
The amount of a radioactive substance remaining after a given amount of time t is given
by the following formula:
62.
A eltln A0
63.
in which A is the amount remaining after time t, variable A0 is the original amount of the
substance, and l is the radioactive decay constant.
64.
65.
61. How much plutonium 239 will remain after 50,000 years if 24 kg was originally
stored? Plutonium 239 has a radioactive decay constant of 0.000029.
66.
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 that
15 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 that
20 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? What
is the difference?
66. How is the pH factor relevant to your selection of a hair care product?
Answers
9. 3 logb x 2 logb y
15. log x 2 log y 21. log2
x2
y
3. log4 x log4 3
11. 2 log4 y 1
log z
2
23. logb x 1y
5. 2 log3 a
1
log4 x
2
7.
1
log5 x
2
13. 2 logb x logb y logb z
1
(log5 x log5 y 2 log5 z)
3
xy2
1y
25. logb
27. log6 3
z
z
17.
19. logb xy
x2y
31. 1.380
33. 0.903
35. 0.151
37. 0.602
B z
39. 0.833
41. 2.833
43. 0.167
45. 7.42, basic
47. 5.61
49. 5610
51. 2 105
53. 0.693
55. 2.303
57. 74
59. 2.930
61. 5.6 kg
63. 60 kg
65.
29. logb
884
3
© 2001 McGraw-Hill Companies
1. logb 5 logb x