56 2. THE LIBRARY OF FUNCTIONS
e next example shows how to deal with a logarithm in an equation by taking a constant to the
power of both sides. To get rid of the logarithm, the constant has to be the base of the logarithm.
Example 2.28 If log
3
.x
2
C x C 1/ D 1:6, find x.
Solution:
log
3
.x
2
C x C 1/ D 1:6
3
log
3
.x
2
CxC1/
D 3
1:6
x
2
C x C 1 D 3
1:6
x
2
C x C 1 3
1:6
D 0
At this point we have converted the problem into a quadratic and apply the quadratic equation:
x D
1 ˙
p
1
2
4 1 .1 3
1:6
/
2 1
x Š 1:7471195 or 2:7471195
˙
We conclude the section on logs and exponentials with the introduction of one of the odder
features of these types of functions, the natural logarithm based on the number e. It takes some
calculus to understand why we use a nutty number like e. For now, just accept that
e Š 2:7182818
For historical reasons there are two logarithm functions that are written without their base.
• ln.x/ is shorthand for log
e
.x/, and
• log.x/ is shorthand for log
10
.x/.
Most of the calculus of exponentials and logs is built around the twin functions y D ln.x/ and
y D e
x
. ese functions are mutual inverses and, in a sense we will understand later, they are the
versions of the log and exponential function that arise naturally from the rest of mathematics.
2.2. POWERS, LOGS, AND EXPONENTIALS 57
PROBLEMS
Problem 2.29 For each of the following expressions, find a simplified version of the the ex-
pression as a power of a single number, as in Example 2.25, where the number was 2.
1.
3
6
3
7
2
2.
1
.
5
3
5
4
/
4
3.
2
4
5
4 2
3
4. 2
2
.
2
2
/
5.
2
2
2
2
6.
7 7
2
7
3
7
4
7
5
7
6
7
7
Problem 2.30 Find the power of a specified by each of these expressions.
1.
5
p
a
15
2.
4
p
a
3
a
5
3.
p
a
5
a
2
3
p
a
4.
a
1=3
a
1=5
8
5. a a
1=2
a
1=3
a
1=4
a
1=5
a
1=6
6. a a
1=2
a
1=3
a
1=4
a
1=5
Problem 2.31 For each of the following functions, find the domain and range of the function.
For some of these, making a plot of the function may help you find the range.
1. f .x/ D
p
x
2
C 1
2. g.x/ D
p
4 x
2
3. h.x/ D
3
p
x C 1
3
4. r.x/ D
p
1 C x
p
1 x
5. s.x/ D
1
x
6. q.x/ D
p
x
3
6x
2
C 11x 6
Problem 2.32 Suppose log
b
.u/ D 2, log
b
.v/ D 1, and log
b
.w/ D 1:2. Compute:
1. log
b
.u v/
2. log
b
.u
6
w/
3. log
b
u
2
v
3
w
1:2
58 2. THE LIBRARY OF FUNCTIONS
4. log
b
.b
4
w
2
/
5. log
b
b
w
u
v
6. log
b
b
4
u
2
Problem 2.33 Solve the following for x.
1. 2
x
D 14
2. 4
x
5 2
x
C 6 D 0
3. .2
x
8/.3
x
9/ D 0
4. 2
x
2
1
D 8
5. 9
x
7 3
x
C 12 D 0
6.
5
p
x
2
C1
D 625
Problem 2.34 For each of the following functions, find the domain and range of the function.
1. f .x/ D
2
x
2
C1
2. g.x/ D
3
1=x
3. h.x/ D
1
p
xC1
4. r.x/ D
1
2
x
5. s.x/ D 2
x
1
6. q.x/ D 2
x
C 3
x
Problem 2.35 Solve the following for x.
1. log
5
.x
2
6x C 8/ D 1
2. 2 log
3
.x/ D log
3
.25/
3. log
2
.1 x/ D 3
4. ln.e
x
C 1/ D 2x
5. log
3
.x C 5/ D 4
6. log
5
.x
2
C 5x C 7/ D 2
Problem 2.36 For each of the following functions, find the domain and range of the function.
Part (f ) may require you to complete a square.
1. f .x/ D ln.x
2
C 1/
2. g.x/ D 5 log
2
.x/
3. h.x/ D log
3
.3
x
C 1/
4. r.x/ D
p
log
2
.x/
2
C 1
5. s.x/ D ln.2 x/
6. q.x/ D log
5
.x
2
C x C 1/
2.2. POWERS, LOGS, AND EXPONENTIALS 59
Problem 2.37 Show, using the algebraic rules for exponents, that
a
b
n
D
b
a
n
Problem 2.38 Show that
n
p
m
p
x D
nm
p
x
Problem 2.39 Look at the Knowledge Boxes for properties of exponential and logarithmic
functions. How are the domain and range of these two types of functions related? Explain.
Problem 2.40 What is the geometric relationship between the graphs of the logarithm and
exponential functions with the same base?
Problem 2.41 Explain in what sense the equation
25
x
4 5
x
C 3 D 0
is a quadratic. Having explained, solve it.
Problem 2.42 Suppose that y D C a
x
. If the points (0,5) and (2,20) are on the graph, then
what are C and a?
Problem 2.43 Suppose that y D C a
x
. If the points (1,6) and (3,54) are on the graph, then
what are C and a?
Problem 2.44 Suppose y D ax C b is a line with positive slope. Find the domain and range
of the function
f .x/ D
p
ax C b
Your answer may be in terms of a and b.
Problem 2.45 Suppose y D ax C b is a line with positive slope. Find the domain and range
of the function
f .x/ D
3
p
ax C b
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