Intermediate Algebra (11th edition)

(d)

Write in exponential form.

Write with the same base.

Power rule for exponents

Set exponents equal.

The solution set is E NOW TRY

1

6 F.

x=

1

6

2 x=

1

3

72 x= 7 1/3

1722 x= 7 1/3

49 x= 237

log 49237 = x

590 CHAPTER 10 Inverse, Exponential, and Logarithmic Functions

Properties of Logarithms

For any positive real number b, with the following are true.

logb b 1 and logb 1  0

bZ 1,

Using Properties of Logarithms

Evaluate each logarithm.

(a) (b)

(c) log 9 1 = 0 logb 1 = 0 (d) log0.2 1 = 0 NOW TRY

log 7 7 = 1 logb b= 1 log 2222 = 1

EXAMPLE 3

OBJECTIVE 4 Define and graph logarithmic functions. Now we define the

logarithmic function with base a.

Logarithmic Function

If aand xare positive numbers, with then

defines the logarithmic function with base a.

g 1 x 2 loga x

aZ1,

Graphing a Logarithmic Function

Graph

By writing in exponential form as we can identify or-

dered pairs that satisfy the equation. It is easier to choose values for yand find the

corresponding values of x. Plotting the points in the table of ordered pairs and

connecting them with a smooth curve gives the graph in FIGURE 10on the next page.

This graph is typical of logarithmic functions with base a 7 1.

y= ƒ 1 x 2 =log 2 x x= 2 y,

ƒ 1 x 2 = log 2 x.

EXAMPLE 4 1 a 712

For any real number b, we know that and for Writing

these statements in logarithmic form gives the following properties of logarithms.

b^1 = b bZ0,b^0 = 1.

NOW TRY
EXERCISE 3
Evaluate each logarithm.

(a)

(b)

(c) log0.1 1

log 8 1

log 10 10

3. (a) 1 (b) 0 (c) 0

NOW TRY
EXERCISE 2
Solve each equation.

(a)

(b)

(c)

(d)log 125235 =x

logx 10 = 2

log3/2 12 x- 12 = 3

log 2 x=- 5

NOW TRY ANSWERS

2. (a) (b)

(c) E 210 F (d) E^19 F

E 321 F E^3516 F

Divide by 2 (which is the same

as multiplying by ).^12