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A. Solving Equations Using the Fundamental Properties of Logarithms

# A. Solving Equations Using the Fundamental Properties of Logarithms

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College Algebra Graphs & Models—

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Section 5.4 Properties of Logarithms

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Fundamental Properties of Logarithms

For any base b 7 0, b 1,

I. logb b ϭ 1, since b1 ϭ b

II. logb 1 ϭ 0, since b0 ϭ 1

III. logb bx ϭ x, since bx ϭ bx

IV. blogb x ϭ x 1x 7 02 , since logb x ϭ logb x

To see the verification of Property IV more clearly, again note that for

y ϭ logb x, b y ϭ x is the exponential form, and substituting logb x for y yields blogb x ϭ x.

Also note that Properties III and IV demonstrate that y ϭ logb x and y ϭ bx are inverse

functions. In common language, “a base-b logarithm undoes a base-b exponential,”

and “a base-b exponential undoes a base-b logarithm.” For f 1x2 ϭ log b x and

f Ϫ1 1x2 ϭ bx, using a composition verifies the inverse relationship just as in Example 5

from Section 5.1:

1 f ‫ ؠ‬f Ϫ1 2 1x2 ϭ f 3 f Ϫ1 1x2 4

1 f Ϫ1 ‫ ؠ‬f 21x2 ϭ f Ϫ1 3 f 1x2 4

ϭ logb bx

ϭx

ϭ blogb x

ϭx

These properties can be used to solve basic equations involving logarithms and exponentials. From the uniqueness property for exponents (page 496), note that if logb x ϭ k

, then blogb x ϭ bk, and we say that we have exponentiated both sides.

EXAMPLE 1

Solving Basic Logarithmic Equations

Solve each equation by applying fundamental properties. Answer in exact form and

approximate form using a calculator (round to 1000ths).

a. ln x ϭ 2

b. Ϫ0.52 ϭ log x

Solution

a. ln x ϭ 2

eln x ϭ e2

x ϭ e2

Ϸ 7.389

given

exponentiate both sides

Property IV, exact form

approximate form

b. Ϫ0.52 ϭ log x

10Ϫ0.52 ϭ 10log x

10Ϫ0.52 ϭ x

0.302 Ϸ x

given

exponentiate both sides

Property IV, exact form

approximate form

Now try Exercises 7 through 10 ᮣ

Note that checking the exact solutions by

substitution is a direct application of Property III

(Figure 5.38).

Also, we observe that exponentiating both sides

of the equation produces the same result as simply

writing the original equation in exponential form, and

the process can be viewed in terms of either

approach.

EXAMPLE 2

Figure 5.38

Solving Basic Exponential Equations

Solve each equation by applying fundamental properties. Answer in exact form and

approximate form using a calculator (round to 1000ths).

a. ex ϭ 167

b. 10 x ϭ 8.223

Solution

a.

ex ϭ 167

ln ex ϭ ln 167

x ϭ ln 167

x Ϸ 5.118

given

use natural log

Property III, exact form

approximate form

b.

10x ϭ 8.223

log 10x ϭ log 8.223

x ϭ log 8.223

x Ϸ 0.915

given

use common log

Property III, exact form

approximate form

Now try Exercises 11 through 14 ᮣ

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CHAPTER 5 Exponential and Logarithmic Functions

Similar to our observations from Example 1, taking

the logarithm of both sides produced the same result

as writing the equation in logarithmic form, and the

process can be viewed in terms of either approach.

Also note that here, checking the exact solution by

substitution is a direct application of Property IV

(Figure 5.39).

If an equation has a single logarithmic or exponential term (base 10 or base e), the equation can be

solved by isolating this term and applying one of the

fundamental properties as in Examples 1 and 2.

EXAMPLE 3

Figure 5.39

Solving Exponential Equations

Solve each equation. Write answers in exact form and approximate form to four

decimal places.

a. 10x Ϫ 29 ϭ 51

b. 3exϩ1 Ϫ 5 ϭ 7

Solution

a. 10x Ϫ 29 ϭ 51 given

10x ϭ 80 add 29

Since the left-hand side is base 10, we apply the common logarithm.

log 10x ϭ log 80

x ϭ log 80

Ϸ 1.9031

take the common log of both sides

Property III (exact form)

approximate form

b. 3exϩ1 Ϫ 5 ϭ 7

given

3exϩ1 ϭ 12 add 5

exϩ1 ϭ 4

divide by 3

Since the left-hand side is base e, we apply the natural logarithm.

ln exϩ1 ϭ ln 4

x ϩ 1 ϭ ln 4

x ϭ ln 4 Ϫ 1

Ϸ 0.3863

take the natural log of both sides

Property III

solve for x (exact form)

approximate form

Now try Exercises 15 through 20 ᮣ

Figure 5.40

As an alternative to using the exact form to

check solutions, we can STO (store) the exact result

in storage location X,T,␪,n (the function variable x) and

simply enter the original equation on the home

screen. The verification for Example 3b is shown in

Figure 5.40.

EXAMPLE 4

Solving Logarithmic Equations

Solve each equation. Write answers in exact form and approximate form to four

decimal places.

a. 2 log 17x2 ϩ 1 ϭ 4

b. Ϫ4 ln 1x ϩ 12 Ϫ 5 ϭ 7

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Section 5.4 Properties of Logarithms

Solution

a. 2 log 17x2 ϩ 1 ϭ 4

2 log 17x2 ϭ 3

3

log 17x2 ϭ

2

3

7x ϭ 102

3

102

7

Ϸ 4.5175

given

subtract 1

divide by 2

exponential form

divide by 7 (exact form)

approximate form

b. Ϫ4 ln 1x ϩ 12 Ϫ 5 ϭ 7

Ϫ4 ln 1x ϩ 12 ϭ 12

ln 1x ϩ 12 ϭ Ϫ3

x ϩ 1 ϭ eϪ3

x ϭ eϪ3 Ϫ 1

Ϸ Ϫ0.9502

given

add 5

divide by Ϫ4

exponential form

subtract 1 (exact form)

approximate form

Now try Exercises 21 through 26

A. You’ve just seen how

we can solve logarithmic

equations using the

fundamental properties of

logarithms

519

Figure 5.41

As with other kinds of equations, solutions to logarithmic and exponential equations can be found

10

using the intersection-of-graphs method or the

zeroes method. For Example 4a and the intersection method, enter Y1 ϭ 2 log17X2 ϩ 1 and

10

Y2 ϭ 4 on the Y= screen (from the domain of 0

the function and the expected result, we know

to set a window that includes only Quadrant I).

Using the 2nd TRACE (CALC) 5:intersect option,

0

we press

three times to identify each curve

and bypass the “Guess” option. The calculator then finds the point of intersection and

prints it at the bottom of the screen, and verifies our calculated result.

ENTER

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

Generally speaking, equation solving involves simplifying the equation, isolating a

variable term on one side, and applying an inverse to solve for the unknown. For logarithmic equations such as log x ϩ log 1x ϩ 32 ϭ 1, we must find a way to combine the

terms on the left, before we can work toward a solution. This requires a further exploration of logarithmic properties.

Due to the close connection between exponents and logarithms, their properties

are very similar. To illustrate, we’ll use terms that can all be written in the form 2x, and

write the equations 8 # 4 ϭ 32, 84 ϭ 2, and 82 ϭ 64 in both exponential form and logarithmic form.

The exponents from a

product are added:

exponential form:

logarithmic form:

The exponents from a

quotient are subtracted:

exponential form:

logarithmic form:

The exponents from a

power are multiplied:

exponential form:

logarithmic form:

23 # 22 ϭ 23ϩ2

log2 18 # 42 ϭ log28 ϩ log24

23

ϭ 23Ϫ2

22

8

log2 a b ϭ log2 8 Ϫ log2 4

4

123 2 2 ϭ 23 2

log2 82 ϭ 2 # log2 8

Each illustration can be generalized and applied with any base b.

#

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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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A. Solving Equations Using the Fundamental Properties of Logarithms

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