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B. The Product, Quotient, and Power Properties of Logarithms

# B. The Product, Quotient, and Power Properties of Logarithms

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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

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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

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

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

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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

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

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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

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B. The Product, Quotient, and Power Properties of Logarithms

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