B. The Product, Quotient, and Power Properties of Logarithms
Tải bản đầy đủ - 0trang
cob19545_ch05_517-527.qxd
11/27/10
12:34 AM
Page 520
College Algebra Graphs & Models—
520
5–42
CHAPTER 5 Exponential and Logarithmic Functions
WORTHY OF NOTE
Properties of Logarithms
For a more detailed verification of
these properties, see Appendix VI.
Given M, N, and b
Product Property
1 are positive real numbers, and any real number p.
Quotient Property
Power Property
M
logb 1MN2 ϭ logb Mϩlogb N
logba b ϭ logb MϪlogb N logb Mp ϭ plogb M
N
The log of a product
The log of a quotient is
The log of a quantity
is a sum of logarithms.
a difference of logarithms.
to a power is the
power times the log
of the quantity.
ᮣ
CAUTION
It’s very important that you read and understand these properties correctly. In particular:
log M
log M
M
(1) log1M ϩ N2 log M ϩ log N, (2) log a b
, and (3)
log M Ϫ log N.
N
log N
log N
For M ϭ 100 and N ϭ 10, statement (1) would indicate log 110 ϭ 2 ϩ 1X, (2) would
indicate that 1 ϭ 21X, and (3) would indicate that 21 ϭ 2 Ϫ 1X.
In the statement of these properties, it’s worth reminding ourselves that the equal
sign “works both ways,” and we have logb M ϩ logb N ϭ logb 1MN2 . These properties
are often used to write a sum or difference of logarithmic terms as a single expression.
EXAMPLE 5
ᮣ
Rewriting Expressions Using Logarithmic Properties
Use the properties of logarithms to write each expression as a single term.
a. log2 7 ϩ log2 5
b. 2 ln x ϩ ln 1x ϩ 62
c. log 1x ϩ 22 Ϫ log x
Solution
ᮣ
a. log2 7 ϩ log2 5 ϭ log2 17 # 52
ϭ log2 35
b. 2 ln x ϩ ln 1x ϩ 62 ϭ ln x2 ϩ ln 1x ϩ 62
ϭ ln 3x2 1x ϩ 62 4
ϭ ln 3x3 ϩ 6x2 4
xϩ2
c. log 1x ϩ 22 Ϫ log x ϭ log a
b
x
product property
simplify
power property
product property
simplify
quotient property
Now try Exercises 27 through 42 ᮣ
We can verify that these properties produce equivalent results by entering the original
equation as Y1, the result after applying the properties as Y2, and viewing the results of
various inputs on a TABLE screen. Results for Example 5b (power and product properties) and 5c (quotient property) are shown in Figures 5.42 and 5.43 respectively.
Figure 5.42
2 ln x ϩ ln1x ϩ 62 ϭ ln1x3 ϩ 6x2 2
Figure 5.43
log 1x ϩ 22 Ϫ log x ϭ log 1 x
ϩ 2
x 2
cob19545_ch05_517-527.qxd
11/27/10
12:35 AM
Page 521
College Algebra Graphs & Models—
5–43
Section 5.4 Properties of Logarithms
EXAMPLE 6
ᮣ
521
Rewriting Logarithmic Expressions Using the Power Property
Use the power property of logarithms to rewrite each term as a product.
a. ln 5x
b. log 32xϩ2
c. log 1x
Solution
a. ln 5x ϭ x ln 5
ᮣ
power property
b. log 32
ϭ 1x ϩ 22 log 32
1
c. log 1x ϭ log x2
1
ϭ log x
2
xϩ2
power property
write radical using a rational exponent
power property
Now try Exercises 43 through 50 ᮣ
For examples of how these properties are used in context, see Exercises 73
and 74.
ᮣ
CAUTION
Note from Example 6b that parentheses must be used whenever the exponent is a sum
or difference. There is a huge difference between 1x ϩ 22 log 32 and x ϩ 2 log 32.
Examples 5 and 6 illustrate how the properties of logarithms are used to consolidate
logarithmic terms, primarily in preparation for equation solving. In other cases, the
properties are used to rewrite or expand logarithmic expressions, so that certain other
procedures can be applied more easily. Example 7 actually lays the foundation for
more advanced mathematical work.
EXAMPLE 7
ᮣ
Rewriting Expressions Using Logarithmic Properties
Use the properties of logarithms to write the following expressions as sums or
differences of simple logarithmic terms.
x
a. log(x2z)
b. ln
Ax ϩ 5
Solution
ᮣ
a. log 1x2z2 ϭ log x2 ϩ log z
ϭ 2 log x ϩ log z
1
2
b. ln
x
x
b
ϭ ln a
xϩ5
Ax ϩ 5
x
1
b
ϭ lna
2
xϩ5
1
ϭ 3 ln x Ϫ ln1x ϩ 52 4
2
product property
power property
write radical using
a rational exponent
power property
quotient property
Now try Exercises 51 through 58 ᮣ
B. You’ve just seen how
we can apply the product,
quotient, and power properties
of logarithms
As you begin working with applications of logarithmic properties, it may help to have
them written on a separate note card. This will enable you to compare each step and
property as they are applied, remembering that “M” and “N” can represent any positive
number or any positive, real-valued expression (see Reinforcing Basic Concepts on
page 527). For Example 7(a) we then have
log1x
z2 ϭ log M
x2 ϩ log Nz
M2 N
cob19545_ch05_517-527.qxd
11/27/10
12:35 AM
Page 522
College Algebra Graphs & Models—
522
5–44
CHAPTER 5 Exponential and Logarithmic Functions
C. The Change-of-Base Formula
Although base-10 and base-e logarithms dominate the mathematical landscape, there are
many practical applications of other bases. Fortunately, a formula exists that will convert
any given base into either base 10 or base e. It’s called the change-of-base formula.
Change-of-Base Formula
For the positive real numbers M, a, and b, with a, b
log M
ln M
logb M ϭ
logb M ϭ
log b
ln b
base 10
1,
logb M ϭ
base e
loga M
loga b
arbitrary base a
Proof of the Change-of-Base Formula
For y ϭ logb M, we have b y ϭ M in exponential form. It follows that
loga 1b y 2 ϭ loga M
y loga b ϭ loga M
loga M
loga b
loga M
logb M ϭ
loga b
yϭ
EXAMPLE 8
ᮣ
take base-a logarithm of both sides
power property of logarithms
divide by loga b
substitute logb M for y
Using the Change-of-Base Formula to Evaluate Expressions
Find the value of each expression using the change-of-base formula. Answer in
exact form and approximate form using nine digits, then verify the result using the
original base. Note that either base 10 or base e can be used.
b. log5 3.6
a. log3 29
Solution
ᮣ
log 29
log 3
Ϸ 3.065044752
a. log3 29 ϭ
Check: 33.065044752 ϭ 29 ✓
ln 3.6
ln 5
Ϸ 0.795888947
b. log5 3.6 ϭ
Check: 50.795888947 ϭ 3.6 ✓
Now try Exercises 59 through 66 ᮣ
C. You’ve just seen how
we can apply the changeof-base formula
The change-of-base formula can also be used to study and graph logarithmic functions of any base. For y ϭ logb x, the right-hand expression is simply rewritten using
log x
the formula and the equivalent function is y ϭ
. The new function can then be
log b
evaluated as in Example 8, or used to study the graph of y ϭ logb x for any base b. See
Exercises 67 through 70.
D. Solving Applications of Logarithms
We end this section with one additional application of logarithms. For all living things,
the concentration of hydrogen ions in a solution plays an important role as their presence or absence alters the environment of other molecules in the solution. This can
dramatically affect the functionality of the solution, or the ability of an organism to
survive. The concentration of hydrogen ions (in moles per liter) is commonly
expressed in terms of what is called the pH scale. A low pH number corresponds to
high hydrogen ion concentration, and the solution is said to be acidic. A high pH
corresponds to low hydrogen ion concentration, and the solution is said to be basic.
Since the range of values is so large, the pH scale is logarithmic, meaning each unit
change in the pH scale represents a 10-fold increase or decrease in the concentration of
cob19545_ch05_517-527.qxd
11/27/10
12:35 AM
Page 523
College Algebra Graphs & Models—
5–45
Section 5.4 Properties of Logarithms
523
hydrogen ions. For example, tomato juice, with a pH level of 2, is 10 times more acidic
than orange juice, with a pH level of 3, and 100 times more acidic than grape juice
(pH ϭ 4; the lower the pH number, the higher the ion concentration). The pH values
range from 0 to 14, with pure water at pH ϭ 7 being deemed “neutral” (neither basic
nor acidic). Measuring pH levels plays an important role in biology, chemistry, food
science, environmental science, medicine, oceanography, personal care products, and
many other areas. The number of hydrogen atoms is usually represented by the term
Hϩ, with the pH number defined as pH ϭ Ϫlog 3H ϩ 4
EXAMPLE 9A
ᮣ
The Concentration of Hydrogen Atoms in Ocean Water
Ocean water has a pH number of near 7.9. What is the concentration of hydrogen
ions? Write the result in scientific notation.
Solution
ᮣ
Begin with the basic formula and work from there.
pH ϭ Ϫlog 3 H ϩ 4
7.9 ϭ Ϫlog 3H ϩ 4
Ϫ7.9 ϭ log 3H ϩ 4
10Ϫ7.9 ϭ H ϩ
1.26 ϫ 10Ϫ8 ϭ H ϩ
pH formula
substitute 7.9 for pH
multiply by Ϫ1
exponential form
result
The hydrogen ion concentration in ocean water is 1.26 ϫ 10Ϫ8 moles/liter.
EXAMPLE 9B
ᮣ
Finding the pH Level of an Apple
The concentration of hydrogen ions in an everyday apple is very near 7.94 ϫ 10Ϫ4.
What is the pH level of an apple?
Solution
D. You’ve just seen how
we can solve applications of
logarithms
pH ϭ Ϫlog 3 H ϩ 4
ϭ Ϫlog 37.94 ϫ 10Ϫ4 4
Ϸ 3.1
ᮣ
pH formula
substitute 7.94 ϫ 10Ϫ4 for Hϩ
result
An apple has a pH level near 3.1.
Now try Exercises 73 through 78 ᮣ
5.4 EXERCISES
ᮣ
CONCEPTS AND VOCABULARY
Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed.
1. For eϪ0.02xϩ1 ϭ 10, the solution process is most
efficient if we apply a base ______ logarithm to
both sides.
4. The statement loge 10 ϭ
2. To solve ln 2x Ϫ ln1x ϩ 32 ϭ 0, we can combine
terms using the ________ property, or add
ln1x ϩ 32 to both sides and use the ________
property.
5. Use all factor pairs of 36 to illustrate the product
property of logarithms. For example, since
36 ϭ 4 и 9, is log(4 и 9) ϭ log 4 ϩ log 9?
3. Since logarithmic functions are not defined for all
real numbers, we should check all “solutions” for
________ roots.
log 10
is an example of
log e
the ________ -of- ________ formula.
6. Use integer divisors of 24 to illustrate the
quotient property of logarithms. For
example, since 12 ϭ 24
2 , is
2
ϭ
log
24 Ϫ log 2?
log 12 ϭ log 1 24
2