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10.6
< 10.6 Objectives >
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Properties of Logarithms
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
© The McGraw-Hill Companies. All Rights Reserved.
The Streeter/Hutchison Series in Mathematics
Elementary and Intermediate Algebra
In this section we develop and use the properties of logarithms. These properties are
applied in a variety of areas that lead to exponential or logarithmic equations.
Since a logarithm is 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.
Property
Properties of
Logarithms
NOTE
The inverse “undoes” what f
does to x.
For b 0 and b 1,
1. logb b 1
Since b1 b
2. logb 1 0
Since b0 1
We know that the logarithmic function y logb x and the exponential function y b x
are inverses of each other. So, for f (x) b x, we have f 1(x) logb x.
For any one-to-one function f,
and
f 1( f (x)) x
for any x in domain of f
f( f 1(x)) x
for any x in domain of f 1
Since f (x) b x is a one-to-one function, we can apply these results to the case where
f(x) b x
and
f 1(x) logb x
to derive some additional properties.
Property
Properties of
Logarithms
3. logb b x x
4. blogbx x
for x 0
Since logarithms are exponents, we can again turn to the familiar exponent rules
to derive some further properties of logarithms.
We know that
log b M x
if and only if
bx M
and log b N y
if and only if
by N
Then
M N b x b y b xy
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From this last equation we see that x y is the power to which we must raise b to get
the product MN. In logarithmic form, that becomes
log b MN x y
Now, since x log b M and y logb N, we can substitute and write
log b MN log b M logb N
This is the first of the basic logarithmic properties presented here. The remaining
properties may all be proved by arguments similar to those presented above.
Property
Properties of Logarithms
Product Property
logb MN logb M logb N
NOTE
Quotient Property
In all cases, M, N 0, b 0,
b 1, and p 0.
M
logb logb M logb N
N
Power Property
Example 1
< Objective 1 >
RECALL
兹a
苶 a1兾2
Using the Properties of Logarithms
Use the properties of logarithms to expand each expression.
(a) logb xy logb x logb y
xy
(b) logb logb xy logb z
z
logb x logb y logb z
Product property
(c) log10 x y log10 x log10 y
Product property
2 3
2
3
2 log10 x 3 log10 y
(d) logb
冪莦y log 冢y冣
x
x
Quotient property
Product property
Power property
1兾2
b
Definition of exponent
1
x
logb 2
y
Power property
1
(logb x logb y)
2
Quotient property
Check Yourself 1
Expand each expression, using the properties of logarithms.
(a) logb x2y3z
(b) log10
冪—莦z—
xy
The Streeter/Hutchison Series in Mathematics
c
© The McGraw-Hill Companies. All Rights Reserved.
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.
Elementary and Intermediate Algebra
logb M p p logb M
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Properties of Logarithms
SECTION 10.6
1053
In some cases, we reverse the process and use the properties to write a single
logarithm, given a sum or difference of logarithmic expressions.
c
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) 5 log10 x 2 log10 y log10 z
log10 x5y2 log10 z
x5y2
log10 z
1
(c) (log2 x log2 y)
2
冢
© The McGraw-Hill Companies. All Rights Reserved.
The Streeter/Hutchison Series in Mathematics
Elementary and Intermediate Algebra
1
x
log2 2
y
冣
冢冣
x
log2 y
log2
Quotient property
1兾2
Power property
冪莦y
x
Check Yourself 2
Write each expression as a single logarithm with coefficient 1.
1
(a) 3 logb x 2 logb y 2 logb z (b) ——(2 log2 x log2 y)
3
Example 3 illustrates the basic concept of the use of logarithms as a computational aid.
c
Example 3
< Objective 2 >
Approximating Logarithms Using Properties
Suppose log10 2 0.301 and log10 3 0.477. Evaluate, as indicated.
> Calculator
(a) log10 6
Since 6 2 3,
NOTES
We wrote 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 only
approximates 2. Verify this
with your calculator.
log10 6 log10 (2 3)
log10 2 log10 3
0.301 0.477
0.778
(b) log10 18
Since 18 2 3 3,
log10 18 log10 (2 3 3)
log10 2 log10 3 log10 3
1.255
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Exponential and Logarithmic Functions
1
(c) log10 9
1
1
Since ,
9
32
1
1
log10 log10 2
9
3
log10 1 log10 32
0 2 log10 3
NOTE
Verify each answer with your
calculator.
logb 1 0 for any base b.
0.954
(d) log10 16
Since 16 24,
log10 16 log10 24 4 log10 2
1.204
(e) log10 兹3苶
Check Yourself 3
Given the values for log10 2 and log10 3, evaluate as indicated.
(a) log10 12
(b) log10 27
(c) log10 兹2
苶
3
When “log” is written without a base, we always assume the base is 10. The LOG
key on your calculator is the log base 10 function. To find log1016, for example, press
LOG 16 ) ENTER . The result should be 1.204, to the nearest thousandth. There
are in fact two logarithm functions built into your graphing calculator, both of which
are frequently used in mathematics.
Logarithms to base 10
Logarithms to base e
Of course, logarithms to base 10 are 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
Definition
The Common
Logarithm, log
log N
means
log10 N
© The McGraw-Hill Companies. All Rights Reserved.
1
log10 兹3苶 log10 31兾2 log10 3
2
0.239
The Streeter/Hutchison Series in Mathematics
Elementary and Intermediate Algebra
Since 兹3苶 31兾2,
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Properties of Logarithms
SECTION 10.6
1055
The table shows the common logarithms for various powers of 10.
NOTE
When no base for a log is
written, it is assumed to be 10.
c
Example 4
> Calculator
Exponential Form
Logarithmic Form
103 1,000
102 100
101 10
100 1
101 0.1
102 0.01
103 0.001
log 1,000
3
log 100
2
log
10
1
log
1
0
log
0.1 1
log
0.01 2
log
0.001 3
Approximating Logarithms with a Calculator
Verify each with a calculator.
(a) log 4.8 0.681
(b) log 48 1.681
© The McGraw-Hill Companies. All Rights Reserved.
The Streeter/Hutchison Series in Mathematics
Elementary and Intermediate Algebra
NOTE
The number 4.8 lies between
1 and 10, so log 4.8 lies
between 0 and 1.
(c) log 480 2.681
(d) log 4,800 3.681
(e) log 0.48 0.319
Check Yourself 4
NOTES
480 4.8 102
and
Use your calculator to evaluate each logarithm, rounded to three
decimal places.
(a) log 2.3
(d) log 2,300
log (4.8 102)
(b) log 23
(e) log 0.23
(c) log 230
(f) log 0.023
log 4.8 log 102
log 4.8 2
2 log 4.8
The value of log 0.48 is really
1 0.681. Your calculator
combines the signed numbers.
Now we 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
a solution is acidic or basic.
The pH of a solution is defined as
pH log [H]
where [H] is the hydrogen ion concentration, in moles per liter (mol/L), in the solution.
c
Example 5
< Objective 3 >
A Chemistry Application
Find the pH of each substance. Determine whether each is a base or an acid.
(a) Rainwater: [H] 1.6 107
NOTES
A solution with pH 7 is
neutral. It is acidic if the pH
is less than 7 and basic if the
pH is greater than 7.
In general, logb b x x, so
log 107 7.
From the definition,
pH log [H]
log (1.6 107)
(log1.6 log107)
艐 [0.204 (7)]
艐 (6.796) 6.796
Rain is slightly acidic.
Use the product rule.
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Exponential and Logarithmic Functions
(b) Household ammonia: [H] 2.3 108
pH log (2.3 108)
(log 2.3 log108)
艐 [0.362 (8)]
艐 7.638
Ammonia is slightly basic.
(c) Vinegar: [H] 2.9 103
pH log (2.9 103)
(log 2.9 log 103)
艐 2.538
Vinegar is very acidic.
Check Yourself 5
c
Example 6
Solving a Logarithmic Equation
Suppose that log x 2.1567. We want to find a number x whose logarithm is 2.1567.
Rewriting in exponential form,
< Objective 4 >
> Calculator
log10 x 2.1567 is equivalent to 102.1567 x
On your graphing calculator, note that the inverse function for LOG is [10x]. So, you
can either press
2nd [10x ] 2.1567 )
ENTER
or, you can directly type
10 ^ 2.1567 ENTER
Both give the result 143.450, rounded to the nearest thousandth, sometimes called the
antilogarithm of 2.1567.
It is important to keep in mind that y log x and y 10x are inverse functions.
Check Yourself 6
In each case, find x to the nearest thousandth.
(a) log x 0.828
(b) log x 1.828
(c) log x 2.828
(d) log x 0.172
Now we return to a chemistry application that requires us to find an antilogarithm.
The Streeter/Hutchison Series in Mathematics
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.
© The McGraw-Hill Companies. All Rights Reserved.
(a) Orange juice: [H] 6.8 105
(b) Drain cleaner: [H] 5.2 1013
Elementary and Intermediate Algebra
Find the pH for each solution. Are they acidic or basic?
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Properties of Logarithms
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Example 7
> Calculator
SECTION 10.6
1057
A Chemistry 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
NOTE
© The McGraw-Hill Companies. All Rights Reserved.
[H] 6.3 107
Check Yourself 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
type of logarithm uses the number e as a base, and we call logarithms to base e natural
logarithms. As with common logarithms, a convenient notation has developed.
Definition
The Natural
Logarithm, ln
The natural logarithm is a logarithm to base e, and it is denoted ln x, where
In x loge x
The Streeter/Hutchison Series in Mathematics
Elementary and Intermediate Algebra
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.
The desired value for [H] is the antilogarithm of 6.2. To find [H], type
2nd [10x ] (-) 6.2 ) ENTER .
The result is 0.00000063, and we can write
The restrictions on the domain of the natural logarithmic function
are the same as before. The function is defined only if x 0.
Since y ln x means y loge x, we can easily convert this to ey x, which leads us
directly to these facts.
ln1 0
ln e 1
ln e2 2
ln e3 3
ln e5 5
Because e0 1
Because e1 e
其
Because ln ex x
We want to emphasize the inverse relationship that exists between logarithmic functions and exponential functions.
Property
Inverse Functions
For any base b,
logb bx x (for all real x)
blogb x x (for x 0)
So, for common logarithms,
log 10x x
10log x x
And, for natural logarithms,
ln ex x
eln x x
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Exponential and Logarithmic Functions
Example 8
Using the Property of Inverses
Simplify.
NOTE
Each of these can be easily
confirmed on a calculator.
But you should learn to
quickly recognize these
forms.
(a)
(b)
(c)
(d)
log 108 8
ln e6 6
10log 7 7
eln 4 4
Check Yourself 8
Simplify.
(a) ln e1.2
Approximating Logarithms with a Calculator
To evaluate natural logarithms, we use a calculator. To find the value of ln 2, use the
sequence
ln 2 )
NOTE
ENTER
The result is 0.693 (to three decimal places).
Check Yourself 9
Use a calculator to evaluate each logarithm. Round to the nearest
thousandth.
(a) ln 3
(b) ln 6
(c) ln 4
(d) ln 兹3
苶
Of course, the properties of logarithms are applied in the same way, no matter
what the base.
c
Example 10
Approximating Logarithms Using Properties
If ln 2 0.693 and ln 3 1.099, evaluate each logarithm.
RECALL
logb MN logb M logb N
logb Mp p logb M
(a) ln 6 ln (2 3) ln 2 ln 3 1.792
(b) ln 4 ln 22 2 ln 2 1.386
1
(c) ln 兹3
苶 ln 31兾2 ln 3 0.550
2
Verify these results with your calculator.
Check Yourself 10
Use ln 2 0.693 and ln 3 1.099 to evaluate each logarithm.
(a) ln 12
(b) ln 27
It may also be necessary to find x, given ln x. The key here is to remember that
y ln x and y ex are inverse functions.
Elementary and Intermediate Algebra
> Calculator
(d) eln 3.7
The Streeter/Hutchison Series in Mathematics
Example 9
(c) log 105
© The McGraw-Hill Companies. All Rights Reserved.
c
(b) 10log 4.5
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Properties of Logarithms
c
Example 11
SECTION 10.6
1059
Solving a Logarithmic Equation
Suppose that ln x = 4.1685. We want to find a number x whose logarithm, base e, is
4.1685. Rewriting in exponential form,
ln x 4.1685 is equivalent to e4.1685 x
On your graphing calculator, note that the inverse function for LN is [ex ]. So, you can
either press
2nd [ex ] 4.1685 )
ENTER
or, you can directly type
2nd ^ 4.1685 ENTER
Both give the result 64.618, rounded to the nearest thousandth.
Check Yourself 11
In each case, find x to the nearest thousandth.
© The McGraw-Hill Companies. All Rights Reserved.
The Streeter/Hutchison Series in Mathematics
Elementary and Intermediate Algebra
(a) ln x 2.065
(b) ln x 2.065
(c) ln x 7.293
The natural logarithm function plays an important role in both theoretical and applied
mathematics. Example 12 illustrates just one of the many applications of this function.
c
Example 12
A Learning Curve Application
A class of students took a 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
RECALL
We read S(t) as “S of t .”
S(t) 76 5 ln (t 1)
S
(a) Find the score after 10 weeks.
80
S(10) 76 5 ln (10 1)
76 5 ln 11 ⬇ 64
60
40
(b) Find the score after 20 weeks.
20
t
10
20
30
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 10.4, exercise 68.
S(20) 76 5 ln (20 1) ⬇ 61
(c) Find the score after 30 weeks.
S(30) 76 5 ln (30 1) ⬇ 59
Check Yourself 12
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 rounded to the nearest tenth after
(a) 3 months
(b) 6 months
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Exponential and Logarithmic Functions
We conclude this section with one final property of logarithms. This property
allows 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.
Suppose we want to find log2 5. This means we want to find the power to which 2
should be raised to produce 5. Now, if there were a log base 2 function (log2 x) on the
calculator, we could obtain this directly. But since there is not, we must take another
approach. If we write
log2 5 x
then we have 2x 5.
Taking the logarithm to base 10 of both sides of the equation yields
log 2x log 5
or
This says 22.322 ⬇ 5.
log 5
x 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 2.322.
log 5
Since x log2 5 and x , then
log 2
log 5
log2 5 log 2
Before leaving this, note that when we took the logarithm (base 10) of both sides,
we could also have taken the logarithm, base e, of both sides.
2x 5
ln 2x ln 5
x ln 2 ln 5
x
ln 5
⬇ 2.322
ln 2
So, log2 5 loge 5
log10 5
.
loge 2
log10 2
Generalizing our result gives us the change-of-base formula.
Property
Change-of-Base
Formula
For positive real numbers a and x,
logb x
loga x logb a
The logarithm on the left side has base a while the logarithms on the right side
have base b. This allows us to calculate the logarithm to base a of any positive number, using the corresponding logarithms to base b (or any other base), as Example 13
illustrates.
Elementary and Intermediate Algebra
NOTE
Now, dividing both sides of this equation by log 2 gives
The Streeter/Hutchison Series in Mathematics
Do not cancel the logs.
Use the power property of logarithms.
© The McGraw-Hill Companies. All Rights Reserved.
>CAUTION
x log 2 log 5
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Example 13
> Calculator
NOTES
We wrote log10 15 rather than
log 15 to emphasize the
change-of-base formula.
log5 5 1 and log5 25 2, so
the result for log5 15 must be
between 1 and 2.
We could choose base e so
ln 15
that log515 instead.
ln 5
SECTION 10.6
1061
Using the Change-of-Base Formula
Find log5 15.
From the change-of-base formula with a 5 and b 10,
log 0 15
log5 15 1
log10 5
1.683
The graphing calculator sequence for the above computation is
log 15 )
log 5 )
ENTER
The result is 1.683, rounded to the nearest thousandth.
Check Yourself 13
Use the change-of-base formula to find log8 32.
© The McGraw-Hill Companies. All Rights Reserved.
The Streeter/Hutchison Series in Mathematics
Elementary and Intermediate Algebra
>CAUTION
A couple of cautions are in order.
1. We cannot “cancel” logs. There is the temptation to write
Remember to close the
parentheses in the numerator
when entering these
expressions into a calculator.
log 15
15
3
log 5
5
This is quite wrong!
2. There is also the temptation to write
log 15
log 15 log 5
log 5
This is also quite wrong.
15
log 15 log 5 (the Quotient Property), but this is very
It is true that log
5
log 15
different from
. Be sure you note the difference.
log 5
冢 冣
Graphing Calculator Option
Applying Logarithmic Regression
A general form of logarithmic functions is available as a regression model in your
graphing calculator: y a b ln x.
Suppose we have collected some data that suggest a pattern of logarithmic growth.
A sample scatterplot that exhibits this is:
We notice relatively rapid growth for small values of x, followed by growth that seems to
be slowing. Look at the data, which show how the time (in seconds) for a dropped tennis
ball to complete its third bounce varies according to the height (in inches) of the drop.
Page 1062
Exponential and Logarithmic Functions
Height of drop (in.)
Time to third bounce (s)
40
45
50
55
60
1.75
1.87
1.99
2.07
2.12
We plot the data, making a scatterplot:
Clear data lists [L1] and [L2]: STAT 4:ClrList 2nd [L1] , 2nd [L2] ENTER .
Enter the data into [L1] and [L2]: STAT 1:Edit, and type in the numbers. Exit the data
editor: 2nd [QUIT].
Make the scatterplot: 2nd [STAT PLOT] ENTER ; press “On”; for “Type” select the
first icon; “Xlist” should say [L1] and “Ylist” should say [L2] ; for “Mark” choose the
first symbol; press Y= and delete (or turn off) any existing equations; press ZOOM
9:ZoomStat. (To improve the scaling, go to WINDOW and choose appropriate numbers
for Xscl and Yscl. Then GRAPH .)
To find the “best fitting” logarithmic function: STAT CALC 9:LnReg 2nd [L1]
, 2nd [L2] ENTER . We have, accurate to four decimal places,
y ⫽ ⫺1.6872 ⫹ 0.9347 ln x
To view the graph of this function on the scatterplot, enter its equation on the Y=
screen and press GRAPH .
Graphing Calculator Check
The table shows the systolic blood pressure p (in mm of Hg) for children of
varying weights w (in pounds). Using your graphing calculator, apply logarithmic regression to fit a logarithmic function to these data. Round coefficients
accurate to four decimal places.
Weight, w
44
61
81
113
131
Blood pressure, p
91
98
103
110
112
ANSWER
p ⫽ 17.9243 ⫹ 19.3850 ln w
Elementary and Intermediate Algebra
CHAPTER 10
12:23 PM
The Streeter/Hutchison Series in Mathematics
1062
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© The McGraw-Hill Companies. All Rights Reserved.
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Properties of Logarithms
SECTION 10.6
1063
Check Yourself ANSWERS
1
1. (a) 2 logb x 3 logb y logb z; (b) (log10 x log10 y log10 z)
2
冪莦
2
x 3y2
3 x
2. (a) logb ;
(b) log2 3. (a) 1.079; (b) 1.431; (c) 0.100
2
y
z
4. (a) 0.362; (b) 1.362; (c) 2.362; (d) 3.362; (e) 0.638; (f) 1.638
5. (a) 4.167, acidic; (b) 12.284, basic
6. (a) 6.730; (b) 67.298; (c) 672.977;
(d) 0.673
7. [H] 1.6 108
8. (a) 1.2; (b) 4.5; (c) 5; (d) 3.7
9. (a) 1.099; (b) 1.792; (c) 1.386; (d) 0.549
10. (a) 2.485; (b) 3.297
11. (a) 7.885; (b) 0.127; (c) 1,469.974
12. (a) 70.5; (b) 65.5
log 32
13. log8 32 ⬇ 1.667
log 8
b
Reading Your Text
SECTION 10.6
© The McGraw-Hill Companies. All Rights Reserved.
The Streeter/Hutchison Series in Mathematics
Elementary and Intermediate Algebra
(a) By definition, a logarithm is an
.
(b) The logarithmic property logb M p p logb M is called the
property.
(c) We call logarithms to the base 10
(d) A solution whose pH 7 is
logarithms.
.
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10.6 exercises
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Page 1064
Basic Skills
|
Challenge Yourself
|
Calculator/Computer
|
Career Applications
|
Above and Beyond
< Objective 1 >
Use the properties of logarithms to expand each expression.
1. logb 5x
2. log3 7x
Name
Section
2
y
3. log6 x
7
4. logb 5. log3 a2
6. log5 y4
7. log5 兹x苶
8. log 兹z苶
9. logb x2y4
10. log7 x3z2
Date
Answers
1.
3
4.
5.
6.
7.
8.
9.
11. log4 y2 兹x苶
12. logb x3 兹z苶
3
10.
The Streeter/Hutchison Series in Mathematics
3.
Elementary and Intermediate Algebra
2.
x2y
z
12.
3
xy
13. logb 14. log5 13.
14.
xy2
兹z苶
15.
15. log 16.
> Videos
x3兹y苶
z
16. log4 2
17.
18.
17. log5
1064
SECTION 10.6
冪莦
3
xy
z2
18. logb
冪莦
4
x2y
z3
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11.
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10.6 exercises
Write each expression as a single logarithm.
19. logb x logb y
20. log5 x log5 y
21. 3 log5 x 2 log5 y
22. 3 logb x logb z
1
23. logb x logb y
2
1
24. logb x 3 logb z
2
25. logb x 2 logb y logb z
26. 2 log5 x (3 log5 y log5 z)
Answers
19.
20.
21.
22.
23.
1
2
27. log6 y 3 log6 z
> Videos
1
3
28. logb x logb y 4 logb z
24.
1
3
© The McGraw-Hill Companies. All Rights Reserved.
The Streeter/Hutchison Series in Mathematics
Elementary and Intermediate Algebra
29. (2 logb x logb y logb z)
1
5
30. (2 log4 x log4 y 3 log4 z)
25.
26.
< Objective 2 >
Given that log 2 0.301 and log 3 0.477, find each logarithm.
31. log 24
27.
32. log 36
28.
33. log 8
34. log 81
35. log 兹2
苶
> Videos
1
4
|
36. log 兹3苶
3
30.
1
27
37. log Basic Skills
29.
38. log Challenge Yourself
| Calculator/Computer | Career Applications
|
41. logm m 0
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
Above and Beyond
Determine whether each statement is true or false.
39. logm xn n logm x
31.
40. (logm x) (logm y) logm xy
x
y
42. logm x logm y logm In exercises 43 to 46, simplify.
43. 10log8.2
44. log 101.3
45. ln e5.8
46. eln 2.6
SECTION 10.6
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10.6 exercises
Estimate each logarithm by “trapping” it between consecutive integers. To estimate
log 4 52, we note that 42 16 and 43 64, so log 4 52 must lie between 2 and 3.
Answers
47. log3 25
48. log5 30
49. log2 70
50. log2 19
51. log 680
52. log 6,800
47.
Without a calculator, use the properties of logarithms to evaluate each expression.
51.
53. log 5 log 2
54. log 25 log 4
52.
55. log3 45 log3 5
56. 10 log4 2
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64.
65.
66.
67.
Basic Skills | Challenge Yourself |
Calculator/Computer
|
Career Applications
|
Above and Beyond
Use your calculator to find each logarithm.
57. log 7.3
58. log 68
59. log 680
60. log 6,800
61. log 0.72
62. log 0.068
63. ln 2
64. ln 3
65. ln 10
66. ln 30
68.
69.
70.
71.
72.
< Objective 4 >
Solve for x. Round to the nearest thousandth.
73.
74.
1066
SECTION 10.6
67. log x 0.749
68. log x 1.749
69. log x 3.749
70. log x 0.251
71. ln x 1.238
72. ln x 3.141
73. ln x 0.786
74. ln x 3.141
The Streeter/Hutchison Series in Mathematics
50.
© The McGraw-Hill Companies. All Rights Reserved.
49.
Elementary and Intermediate Algebra
48.
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10.6 exercises
< Objective 3 >
You are given the hydrogen ion concentration [H] for each solution. Use the formula
pH log [H] to find each pH. Are the solutions acidic or basic?
Answers
75. Blood: [H] 3.8 108
75.
76. Lemon juice: [H] 6.4 103
Given the pH of the solutions, approximate the hydrogen ion concentration [H].
77. Wine: pH 4.7
76.
78. Household ammonia: pH 7.8
> Videos
77.
The average score on a final examination for a group of psychology students, retested
after time t (in weeks), is given by
78.
S 85 8 ln (t 1)
79.
Find the average score on the retests.
© The McGraw-Hill Companies. All Rights Reserved.
The Streeter/Hutchison Series in Mathematics
Elementary and Intermediate Algebra
79. After 3 weeks
80. After 12 weeks
80.
Use the change-of-base formula to approximate each logarithm.
81.
81. log3 25
82. log5 30
> Videos
82.
83. The table shows measurements taken for several trees of the same species.
The measurements were diameter at the base, in centimeters (cm), and
height, in meters (m). Using your graphing calculator, apply logarithmic
regression to fit a logarithmic function to these data. Round coefficients
accurate to four decimal places.
83.
84.
x (diameter)
2.6
4.6
9.8
14.5
15.8
27.0
y (height)
1.93
4.15
11.50
11.85
13.25
15.80
84. The table below shows measurements taken for several trees of the same
species. The measurements were diameter at the base, in centimeters (cm),
and crown width, in meters (m). Using your graphing calculator, apply
logarithmic regression to fit a logarithmic function to these data. Round
coefficients accurate to four decimal places.
x (diameter)
2.6
4.6
9.8
14.5
15.8
27.0
y (crown width)
0.5
1.6
3.6
3.7
4.0
6.5
SECTION 10.6
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10.6 exercises
Basic Skills
|
Challenge Yourself
|
Calculator/Computer
|
Career Applications
|
Above and Beyond
Answers
The amount of a radioactive substance remaining after time t is given by
85.
A eltln A0
86.
chapter
10
> Make the
Connection
where A is the amount remaining after time t, A0 is the original amount of the
substance, and l is the radioactive decay constant. Assume t is measured in years.
87.
85. How much plutonium-239 will remain after 50,000 years if 24 kg was origi-
nally stored? Plutonium-239 has a radioactive decay constant of 0.000029.
88.
chapter
10
> Make the
Connection
89.
86. How much plutonium-241 will remain after 100 years if 52 kg was origi-
90.
nally stored? Plutonium-241 has a radioactive decay constant of 0.053319.
> Make the
91.
87. How much strontium-90 was originally stored if after 56 years it is discov-
ered that 15 kg still remains? Strontium-90 has a radioactive decay constant
of 0.024755.
>
chapter
10
Make the
Connection
88. 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.
>
chapter
10
Make the
Connection
89. Which keys on your calculator are function keys and which are operation
keys? What is the difference?
90. How is the pH factor relevant to your selection of a hair-care product?
91. (a) Use the change-of-base formula to write log3 8 in terms of base-10 loga-
rithms. Then use your calculator to find log3 8 rounded to three decimal
places.
(b) Use the change-of-base formula to write log3 8 in terms of base-e logarithms. Then use your calculator to find log3 8 rounded to three decimal
places.
(c) Compare your answers to parts (a) and (b).
1068
SECTION 10.6
Elementary and Intermediate Algebra
Connection
The Streeter/Hutchison Series in Mathematics
10
© The McGraw-Hill Companies. All Rights Reserved.
chapter
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Page 1069
10.6 exercises
Answers
1. logb 5 logb x
3. log6 x log6 7
9. 2 logb x 4 logb y
1
2
11. 2 log 4 y log 4 x
13. 2 logb x logb y logb z
1
3
17. (log5 x log5 y 2 log5 z)
19. logb xy
x3
y
21. log5 2
x
yz
25. log b 2
27. log 6 3
冪莦
© The McGraw-Hill Companies. All Rights Reserved.
The Streeter/Hutchison Series in Mathematics
Elementary and Intermediate Algebra
91.
1
2
15. log x 2 log y log z
2
兹苶y
3 x y
29. log b z
z
1.380
33. 0.903
35. 0.151
37. 0.602
39. True
False
43. 8.2
45. 5.8
47. Between 2 and 3
Between 6 and 7
51. Between 2 and 3
53. 1
55. 2
0.863
59. 2.833
61. 0.143
63. 0.693
65. 2.303
5.610
69. 5,610.480
71. 3.449
73. 0.456
75. 7.42, basic
2 105
79. 74
81. 2.930
83. y 4.2465 6.2220 ln x
5.6 kg
87. 60 kg
89. Above and Beyond
log 8
ln 8
(a)
, 1.893; (b)
, 1.893; (c) same
log 3
ln 3
23. log b x 兹y苶
31.
41.
49.
57.
67.
77.
85.
1
2
7. log5 x
5. 2 log3 a
SECTION 10.6
1069