Intermediate Algebra (11th edition)

Graphing a Logarithmic Function

Graph

We write in exponential form as then choose values

for yand find the corresponding values of x. Plotting these points and connecting

them with a smooth curve gives the graph in FIGURE 11. This graph is typical of loga-

rithmic functions with base 0 6 a 6 1.

x= A^12 B

y

y= g 1 x 2 =log1/2 x ,

g 1 x 2 = log1/2 x.

EXAMPLE 5 106 a 612

SECTION 10.3 Logarithmic Functions 591

• 1

y

0

2

4

x

y

• 2
  
  
  • 4
  
  
  • 2
  
  
  
  

24

1

2

(^14) – 2

f f (x) log 22 x

1

2

4

0

1

2

x 22 y

FIGURE 10 NOW TRY

Logarithmic function

with base

Domain:

Range:

The function is one-to-one,

and its graph rises from

left to right.

1 - q, q 2

1 0, q 2

a> 1

Logarithmic function

with base

Domain:

Range:

The function is one-to-one,

and its graph falls from

left to right.

1 - q, q 2

1 0, q 2

0 <a< 1

1

y

0

2

4

x

y

• 2
  
  
  • 4
  
  
  • 2
  
  
  
  

24

1

2

(^142)

x

1

2

4

0

• 1


• 2

y

x^12

g (x) log1/21/2

FIGURE 11 NOW TRY

Characteristics of the Graph of

1. The graph contains the point

2. The function is one-to-one. When the graph will risefrom left to right,

from the fourth quadrant to the first. (SeeFIGURE 10.) When the

graph will fallfrom left to right, from the first quadrant to the fourth. (See

FIGURE 11.)

3. The graph will approach the y-axis, but never touch it. (The y-axis is an

asymptote.)

4. The domain is 1 0, q 2 ,and the range is 1 - q, q 2.

06 a 6 1,

a 7 1,

1 1, 0 2.

g 1 x 2 loga x

NOW TRY
EXERCISE 4
Graph ƒ 1 x 2 =log 6 x.

NOW TRY ANSWERS
4.

x

y

0

1

16

f(x) = log 6 x

NOW TRY
EXERCISE 5
Graph g 1 x 2 =log1/4 x.

5.

x

y

0

11 4

gg((xx) = log) = log1/41/4xx

NOTE See the box titled “Characteristics of the Graph of ” on page 582.

Below we give a similar set of characteristics for the graph of Com-

pare the four characteristics one by one to see how the concepts of inverse functions,

introduced in Section 10.1,are illustrated by these two classes of functions.

g 1 x 2 =loga x.

ƒ 1 x 2 = ax

Be careful to write

the x- and y-values

in the correct

order.