Chapter Eighteen — Logarithmic Functions

The Humongous Book of Algebra Problems

412

18.32 Solve the equation 9

x

= 13 using the change of base formula and round the

answer to the thousandths place.

Express the exponential equation as a logarithmic equation and solve.

18.33 Solve the equation 7

2x

+ 1 = 6 using the change of base formula and round the

answer to the thousandths place.

Isolate 7

2x

on the left side of the equation.

Rewrite the exponential equation as a logarithmic equation.

log

7

5 = 2x

Calculate log

7

5 using the change of base formula.

Multiply both sides of the equation by to solve for x.

Logarithmic Properties

Expanding, contracting, and simplifying log expressions

18.34 Expand the expression: ln (2x).

Logarithmic expressions can be rewritten according to three fundamental

properties. Speciﬁcally, a single logarithm can be expressed as multiple

logarithms and vice versa. The ﬁrst property states that the logarithm of a

product can be expressed as the sum of the logarithms of its factors:

log

a

(xy) = log

a

x + log

a

y.

Or divide

both sides by 2

Which are

presented one at a

time in Problems 18.34,

18.36, and 18.38

Chapter Eighteen — Logarithmic Functions

The Humongous Book of Algebra Problems

413

In this example, the logarithm of the product 2x may be expressed as the sum of

the logarithms of its factors, 2 and x.

ln (2x) = ln 2 + ln x

18.35 Demonstrate the logarithmic property presented in Problem 18.34 by verifying

that ln 10 = ln 2 + ln 5.

Because 2(5) = 10, the natural logarithm of the product (10) should equal the

sum of the natural logarithms of the factors (2 and 5). Verify using a calculator.

18.36 Expand the expression: .

A property of logarithms states that the logarithm of a quotient is equal to the

difference of the logarithms of the dividend and divisor: .

18.37 Demonstrate the logarithmic property presented in Problem 18.36 by

verifying that log 4 = log 12 – log 3.

Because , the logarithm of the quotient (4) is equal to the difference of

the logarithms of the dividend (12) and the divisor (3).

Note: Problems 18.38–18.39 refer to the expression log y

3

.

18.38 Expand the logarithmic expression.

A property of logarithms states that the logarithm of a quantity raised to an

exponent n is equal to the product of n and the logarithm of the base:

log

a

x

n

= n(log

a

x).

log y

3

= 3(log y)

There are

no properties for

the log of a sum. For

example, ln (2 + x) ≠

(ln 2)(ln x). You also can’t

“distribute” the letters

“ln.” For example,

ln (x + 2) ≠ ln x + ln 2.

This is not

only true for

natural logs but

it also works for any

base, as long as all the

bases in the expression

match. In other words,

log 10 = log 2 + log 5

and log

12

10 = log

12

2 + log

2

5.

In other

words, the log of a

fraction equals the

log of the numerator

minus the log of the

denominator.

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