10
Exponential and
Logarithmic
Functions
10.1 Algebra and Composition of Functions
10.2 Inverse Functions
10.3 Exponential Functions
10.4 Logarithmic Functions
10.5 Properties of Logarithms
10.6 The Irrational Number e
Problem Recognition Exercises—Logarithmic and
Exponential Forms
10.7 Logarithmic and Exponential Equations
Chapter 10 is devoted to the study exponential and logarithmic functions.
These functions are used to study many naturally occurring phenomena such
as population growth, exponential decay of radioactive matter, and growth of
investments.
The following is a Sudoku puzzle. As you work through this chapter, try
to simplify the expressions or solve the equations in the clues given below.
Use the clues to fill in the boxes labeled a–n. Then fill in the remaining part of
the grid so that every row, every column, and every 2 3 box contains the
digits 1 through 6.
Clues
a b
a. log2 8 b. In e
c. Solution to 3 2x10
9 c d
d. Solution to ln 1x 32 ln 8
e. 1 log 1 f. log2 16x log2 x e f g h
g. Solution to log12 1x 42 1 log12 x
ln 22
i j k
h. i. f1f1 1622
ln 2
j. ln e4 k. log7 75 l m
1 x
l. e0 m. a b 16 n
2
n. log 100
689
,690 Chapter 10 Exponential and Logarithmic Functions
Section 10.1 Algebra and Composition of Functions
Concepts 1. Algebra of Functions
1. Algebra of Functions Addition, subtraction, multiplication, and division can be used to create a new
2. Composition of Functions function from two or more functions. The domain of the new function will be
3. Multiple Operations on the intersection of the domains of the original functions. Finding domains of
Functions functions was first introduced in Section 4.2.
Sum, Difference, Product, and Quotient of Functions
f
Given two functions f and g, the functions f g, f g, f g, and g are defined
as
1f g21x2 f 1x2 g1x2
1f g2 1x2 f 1x2 g1x2
1f g2 1x2 f 1x2 g1x2
f 1x2
a b1x2
f
provided g1x2 0
g g1x2
( f g)(x) |x| 3
y For example, suppose f 1x2 ƒ x ƒ and g1x2 3. Taking the sum of the functions pro-
7
duces a new function denoted by 1 f g2. In this case, 1 f g21x2 ƒ x ƒ 3. Graph-
ically, the y-values of the function 1 f g2 are given by the sum of the corresponding
6
f (2) g(2)
f (x) |x|
y-values of f and g. This is depicted in Figure 10-1. The function 1 f g2 appears in
5
4
g(x) 3
2
red. In particular, notice that 1 f g2122 f 122 g122 2 3 5.
1
x
5 4 3 2 1 1 2 3 4 5
1 Example 1 Adding, Subtracting, and Multiplying Functions
2
3
Given: g1x2 4x h1x2 x2 3x k1x2 2x 2
Figure 10-1
a. Find 1g h2 1x2 and write the domain of 1g h2 in interval notation.
b. Find 1h g2 1x2 and write the domain of 1h g2 in interval notation.
c. Find 1g k2 1x2 and write the domain of 1g k2 in interval notation.
Solution:
a. 1g h2 1x2 g1x2 h1x2
14x2 1x 2 3x2
4x x 2 3x
x2 x The domain is all real numbers 1, 2 .
b. 1h g2 1x2 h1x2 g1x2
1x2 3x2 14x2
x2 3x 4x
x 2 7x The domain is all real numbers 1, 2 .
c. 1g k2 1x2 g1x2 k1x2
14x21 1x 22
4x 1x 2 The domain is 32, 2 because x 2 0
for x 2.
, Section 10.1 Algebra and Composition of Functions 691
Skill Practice Given:
f 1x2 x 1
g 1x2 5x2 x
h1x2 25 x
Perform the indicated operations. Write the domain of the resulting function in in-
terval notation.
1. 1f g21x2 2. 1g f 21x2 3. 1f h21x2
Example 2 Dividing Functions
Given the functions defined by h1x2 x2 3x and k1x2 1x 2, find Qkh R 1x2 and
write the domain of Qkh R in interval notation.
Solution:
1x 2
a b1x2 2
k
h x 3x
To find the domain, we must consider the restrictions on x imposed by the square
root and by the fraction.
• From the numerator we have x 2 0 or, equivalently, x 2.
• From the denominator we have x2 3x 0 or, equivalently, x1x 32 0.
Hence, x 3 and x 0.
Thus, the domain of kh is the set of real numbers [
5 4 32 1 0 1 2 3 4 5
greater than or equal to 2, but not equal to 3 or 0.
Figure 10-2
This is shown graphically in Figure 10-2.
The domain is [2, 3) ´ (3, ).
Skill Practice Given:
f 1x2 1x 1
g 1x2 x2 2x
4. Find a b1x2 and write the domain interval notation.
f
g
2. Composition of Functions
Composition of Functions
The composition of f and g, denoted f g, is defined by the rule
1 f g21x2 f 1g1x2 2 provided that g1x2 is in the domain of f
The composition of g and f, denoted g f, is defined by the rule Skill Practice Answers
1. 5x 2 2x 1; domain: 1, 2
1g f 21x2 g1 f 1x2 2 provided that f 1x2 is in the domain of g 2. 5x 2 1; domain: 1, 2
3. 1x 12 15 x ; domain: 1, 5 4
Note: f g is also read as “f compose g,” and g f is also read as “g compose f.”
1x 1
4. 2 ; domain: 31,02 ´10, 2
x 2x
, 692 Chapter 10 Exponential and Logarithmic Functions
For example, given f 1x2 2x 3 and g1x2 x 5, we have
1f g2 1x2 f 1g1x22
f 1x 52 Substitute g1x2 x 5 into the function f.
21x 52 3
2x 10 3
2x 7
In this composition, the function g is the innermost operation and acts on x first.
Then the output value of function g becomes the domain element of the function
f, as shown in the figure.
g f
x g(x) f(g(x))
Example 3 Composing Functions
Given: f 1x2 x 5, g1x2 x2, and n1x2 1x 2
a. Find 1 f g21x2 and write the domain of 1 f g2 in interval notation.
b. Find 1g f 21x2 and write the domain of 1g f 2 in interval notation.
c. Find 1n f 2 1x2 and write the domain of 1n f 2 in interval notation.
Solution:
a. 1 f g21x2 f 1g1x2 2
f 1x 2 2 Evaluate the function f at x2.
1x 2 2 5 Replace x by x2 in the function f.
TIP: Examples 3(a) and x2 5 The domain is all real numbers 1, 2 .
b. 1g f 21x2 g1 f 1x2 2
3(b) illustrate that the
order in which two
functions are composed g1x 52 Evaluate the function g at 1x 52.
1x 52 Replace x by 1x 52 in function g.
may result in different 2
functions. That is, f g
does not necessarily x2 10x 25 The domain is all real numbers 1, 2 .
c. 1n f 21x2 n1 f 1x22
equal g f.
n1x 52 Evaluate the function n at x 5.
11x 52 2 Replace x by the quantity 1x 52 in function n.
1x 3 The domain is 33, 2 .
Skill Practice Given f 1x2 2x2, g 1x2 x 3, and h 1x2 1x 1,
5. Find 1 f g2 1x2 . Write the domain of 1 f g2 in interval notation.
6. Find 1g f 2 1x2 . Write the domain of 1g f 2 in interval notation.
7. Find 1h g2 1x2 . Write the domain of 1h g2 in interval notation.
Skill Practice Answers
5. 2x 2 12x 18; domain: 1, 2
6. 2x 2 3; domain: 1, 2
7. 1x 2; domain: 32, 2
Exponential and
Logarithmic
Functions
10.1 Algebra and Composition of Functions
10.2 Inverse Functions
10.3 Exponential Functions
10.4 Logarithmic Functions
10.5 Properties of Logarithms
10.6 The Irrational Number e
Problem Recognition Exercises—Logarithmic and
Exponential Forms
10.7 Logarithmic and Exponential Equations
Chapter 10 is devoted to the study exponential and logarithmic functions.
These functions are used to study many naturally occurring phenomena such
as population growth, exponential decay of radioactive matter, and growth of
investments.
The following is a Sudoku puzzle. As you work through this chapter, try
to simplify the expressions or solve the equations in the clues given below.
Use the clues to fill in the boxes labeled a–n. Then fill in the remaining part of
the grid so that every row, every column, and every 2 3 box contains the
digits 1 through 6.
Clues
a b
a. log2 8 b. In e
c. Solution to 3 2x10
9 c d
d. Solution to ln 1x 32 ln 8
e. 1 log 1 f. log2 16x log2 x e f g h
g. Solution to log12 1x 42 1 log12 x
ln 22
i j k
h. i. f1f1 1622
ln 2
j. ln e4 k. log7 75 l m
1 x
l. e0 m. a b 16 n
2
n. log 100
689
,690 Chapter 10 Exponential and Logarithmic Functions
Section 10.1 Algebra and Composition of Functions
Concepts 1. Algebra of Functions
1. Algebra of Functions Addition, subtraction, multiplication, and division can be used to create a new
2. Composition of Functions function from two or more functions. The domain of the new function will be
3. Multiple Operations on the intersection of the domains of the original functions. Finding domains of
Functions functions was first introduced in Section 4.2.
Sum, Difference, Product, and Quotient of Functions
f
Given two functions f and g, the functions f g, f g, f g, and g are defined
as
1f g21x2 f 1x2 g1x2
1f g2 1x2 f 1x2 g1x2
1f g2 1x2 f 1x2 g1x2
f 1x2
a b1x2
f
provided g1x2 0
g g1x2
( f g)(x) |x| 3
y For example, suppose f 1x2 ƒ x ƒ and g1x2 3. Taking the sum of the functions pro-
7
duces a new function denoted by 1 f g2. In this case, 1 f g21x2 ƒ x ƒ 3. Graph-
ically, the y-values of the function 1 f g2 are given by the sum of the corresponding
6
f (2) g(2)
f (x) |x|
y-values of f and g. This is depicted in Figure 10-1. The function 1 f g2 appears in
5
4
g(x) 3
2
red. In particular, notice that 1 f g2122 f 122 g122 2 3 5.
1
x
5 4 3 2 1 1 2 3 4 5
1 Example 1 Adding, Subtracting, and Multiplying Functions
2
3
Given: g1x2 4x h1x2 x2 3x k1x2 2x 2
Figure 10-1
a. Find 1g h2 1x2 and write the domain of 1g h2 in interval notation.
b. Find 1h g2 1x2 and write the domain of 1h g2 in interval notation.
c. Find 1g k2 1x2 and write the domain of 1g k2 in interval notation.
Solution:
a. 1g h2 1x2 g1x2 h1x2
14x2 1x 2 3x2
4x x 2 3x
x2 x The domain is all real numbers 1, 2 .
b. 1h g2 1x2 h1x2 g1x2
1x2 3x2 14x2
x2 3x 4x
x 2 7x The domain is all real numbers 1, 2 .
c. 1g k2 1x2 g1x2 k1x2
14x21 1x 22
4x 1x 2 The domain is 32, 2 because x 2 0
for x 2.
, Section 10.1 Algebra and Composition of Functions 691
Skill Practice Given:
f 1x2 x 1
g 1x2 5x2 x
h1x2 25 x
Perform the indicated operations. Write the domain of the resulting function in in-
terval notation.
1. 1f g21x2 2. 1g f 21x2 3. 1f h21x2
Example 2 Dividing Functions
Given the functions defined by h1x2 x2 3x and k1x2 1x 2, find Qkh R 1x2 and
write the domain of Qkh R in interval notation.
Solution:
1x 2
a b1x2 2
k
h x 3x
To find the domain, we must consider the restrictions on x imposed by the square
root and by the fraction.
• From the numerator we have x 2 0 or, equivalently, x 2.
• From the denominator we have x2 3x 0 or, equivalently, x1x 32 0.
Hence, x 3 and x 0.
Thus, the domain of kh is the set of real numbers [
5 4 32 1 0 1 2 3 4 5
greater than or equal to 2, but not equal to 3 or 0.
Figure 10-2
This is shown graphically in Figure 10-2.
The domain is [2, 3) ´ (3, ).
Skill Practice Given:
f 1x2 1x 1
g 1x2 x2 2x
4. Find a b1x2 and write the domain interval notation.
f
g
2. Composition of Functions
Composition of Functions
The composition of f and g, denoted f g, is defined by the rule
1 f g21x2 f 1g1x2 2 provided that g1x2 is in the domain of f
The composition of g and f, denoted g f, is defined by the rule Skill Practice Answers
1. 5x 2 2x 1; domain: 1, 2
1g f 21x2 g1 f 1x2 2 provided that f 1x2 is in the domain of g 2. 5x 2 1; domain: 1, 2
3. 1x 12 15 x ; domain: 1, 5 4
Note: f g is also read as “f compose g,” and g f is also read as “g compose f.”
1x 1
4. 2 ; domain: 31,02 ´10, 2
x 2x
, 692 Chapter 10 Exponential and Logarithmic Functions
For example, given f 1x2 2x 3 and g1x2 x 5, we have
1f g2 1x2 f 1g1x22
f 1x 52 Substitute g1x2 x 5 into the function f.
21x 52 3
2x 10 3
2x 7
In this composition, the function g is the innermost operation and acts on x first.
Then the output value of function g becomes the domain element of the function
f, as shown in the figure.
g f
x g(x) f(g(x))
Example 3 Composing Functions
Given: f 1x2 x 5, g1x2 x2, and n1x2 1x 2
a. Find 1 f g21x2 and write the domain of 1 f g2 in interval notation.
b. Find 1g f 21x2 and write the domain of 1g f 2 in interval notation.
c. Find 1n f 2 1x2 and write the domain of 1n f 2 in interval notation.
Solution:
a. 1 f g21x2 f 1g1x2 2
f 1x 2 2 Evaluate the function f at x2.
1x 2 2 5 Replace x by x2 in the function f.
TIP: Examples 3(a) and x2 5 The domain is all real numbers 1, 2 .
b. 1g f 21x2 g1 f 1x2 2
3(b) illustrate that the
order in which two
functions are composed g1x 52 Evaluate the function g at 1x 52.
1x 52 Replace x by 1x 52 in function g.
may result in different 2
functions. That is, f g
does not necessarily x2 10x 25 The domain is all real numbers 1, 2 .
c. 1n f 21x2 n1 f 1x22
equal g f.
n1x 52 Evaluate the function n at x 5.
11x 52 2 Replace x by the quantity 1x 52 in function n.
1x 3 The domain is 33, 2 .
Skill Practice Given f 1x2 2x2, g 1x2 x 3, and h 1x2 1x 1,
5. Find 1 f g2 1x2 . Write the domain of 1 f g2 in interval notation.
6. Find 1g f 2 1x2 . Write the domain of 1g f 2 in interval notation.
7. Find 1h g2 1x2 . Write the domain of 1h g2 in interval notation.
Skill Practice Answers
5. 2x 2 12x 18; domain: 1, 2
6. 2x 2 3; domain: 1, 2
7. 1x 2; domain: 32, 2
