Intermediate Algebra (11th edition)

SECTION 10.3 Logarithmic Functions 589

Exponential Form Logarithmic Form

4 -^3 = 641 log 4 641 =- 3

105 =100,000 log 10 100,000= 5

A 51 B-^2 =^25 log1/5^25 =-^2

32 = 9 log 3 9 = 2

means

x=ay.

y=loga x

Converting Between Exponential and Logarithmic Forms

The table shows several pairs of equivalent forms.

EXAMPLE 1

NOW TRY

OBJECTIVE 3 Solve logarithmic equations of the form for a, b,

or k.A logarithmic equationis an equation with a logarithm in at least one term.

loga bk

Solving Logarithmic Equations

Solve each equation.

(a)

By the definition of logarithm, is equivalent to

The solution set is

(b)

Write in exponential form.

Apply the exponent.

Multiply each term by 4.

Subtract 4.

x=-

1

4

12 x=- 3

12 x+ 4 = 1

3 x+ 1 =

1

4

3 x+ 1 = a

1

2

b

2

log1/2 13 x+ 12 = 2

E

1

16 F.

x= 4 -^2 =

1

16

log 4 x=- 2 x= 4 -^2.

log 4 x=- 2

EXAMPLE 2

This is a key step.

CHECK Let

Simplify within parentheses.

✓ Exponential form; true

The solution set is

(c)

Write in exponential form.

x=  23 Take square roots.

x^2 = 3

logx 3 = 2

E-

1

4 F.

a

1

2

b

2

=

1

4

log1/2

1

4

 2

log1/2 a 3 a- x=- 41.

1

4

b + 1 b 2

Be careful here.

• 23 is extraneous.

Only the principalsquare root satisfies the equation since the base must be a positive

number. The solution set is E (^23) F.
NOW TRY
EXERCISE 1
(a)Write in
logarithmic form.
(b)Write in
exponential form.
log 64 4 =^13

63 = 216

NOW TRY ANSWERS

1. (a)
   (b) 64 1/3= 4

log 6 216 = 3

Divide by 12. Write

in lowest terms.