7-5
Attributes of Logarithmic Functions
TEKS FOCUS
VOCABULARY
TEKS (5)(C) Rewrite exponential equations
as their corresponding logarithmic
equations and logarithmic equations as their
corresponding exponential equations.
TEKS (1)(F) Analyze mathematical
relationships to connect and communicate
mathematical ideas.
Additional TEKS (1)(D), (2)(A), (2)(B),
(2)(C)
ĚCommon logarithm – A common
logarithm is a logarithm
with base 10. You can write
a common logarithm log10 x
simply as log x, without showing
the 10.
ĚLogarithmic function – the
inverse of an exponential
function
ĚAnalyze – closely examine
ĚLogarithm – The logarithm
objects, ideas, or relationships to
learn more about their nature
base b of a positive number x is
defined as follows: logb x = y, if
and only if x = by .
ESSENTIAL UNDERSTANDING
The exponential function y = bx is one-to-one, so its inverse x = by is a function.
To express “y as a function of x” for the inverse, write y = log b x.
Key Concept
Logarithm
A logarithm base b of a positive number x satisfies the following definition.
For b 7 0, b ≠ 1, log b x = y if and only if b y = x.
You can read log b x as “log base b of x.” In other words, the logarithm y is the
exponent to which b must be raised to get x.
The exponent y in the expression by is the logarithm in the equation log b x = y. The
base b in by and the base b in log b x are the same. In both, b ≠ 1 and b 7 0.
Since b ≠ 1 and b 7 0, it follows that by 7 0. Since by = x then x 7 0, so log b x is
defined only for x 7 0.
Because y = bx and y = log b x are inverse functions, their compositions map a
number a to itself. In other words, blogba = a for a 7 0 and log b ba = a for all a.
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Problem 1
P
Writing Exponential Equations in Logarithmic Form
What is the logarithmic form of each equation?
To what power do
you raise 10 to get
100?
10 raised to the 2nd
power equals 100.
A 100 = 102
Use the definition of logarithm.
If x = by
then
log b x = y
If 100 = 102
then
log 10 100 = 2
B 81 =
34
Use the definition of logarithm.
If x = by
then
log b x = y
If 81 = 34
then
log 3 81 = 4
Problem
P
bl
2
TEKS Process Standard (1)(F)
Evaluating a Logarithm
Multiple Choice What is the value of log 8 32?
3
5
How can you use
the definition of
logarithm to help
you find the value of
log8 32?
If logb x = y then
x = b y, so to what
power must you raise
8 to get 32?
5
3
log 8 32 = x
32 =
8x
Use the definition of a logarithm to write an exponential equation.
25 = (23)x
Write each side using base 2.
25 = 23x
Power Property of Exponents
5 = 3x
Since the bases are the same, the exponents must be equal.
5
3=x
Solve for x.
Since 8 = 32, then log 8 32 = 53.
The correct answer is B.
Lesson 7-5
5
Write a logarithmic equation.
5
3
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3
Attributes of Logarithmic Functions
Problem 3
P
TEKS Process Standard (1)(F)
Graphing the Inverse of a Function
A digital camera has a button, which enlarges the image on the view screen
10 times for every second the button is pressed. The function f (x) represents the
scale factor by which the image on the view screen has been enlarged when the
button has been pressed for x seconds.
A Find the function f (x) representing the scale factor and its inverse, f −1(x).
Describe and analyze the domain and range of f and f −1 .
The scale factor is an exponential relationship, as the scale factor is multiplied by
10 each second the button is pushed. So, f (x) = 10 x .
To find the inverse of this, switch x and y in the original relation and solve for x.
y = 10x
x=
Original relation
10 y
Switch x and y.
log x = y
f -1(x)
Take the logarithm of each side.
= log x
Because a button cannot be pushed for a negative amount of time, the domain of
f (x) is x Ú 0. This generates the range y Ú 1.
How are the graphs
of functions and their
inverses related?
The graphs of inverse
functions are reflections
of each other across the
line y = x. You can graph
y = log x as the inverse
of y = 10x .
Since f -1 is the inverse of f, use the range of f as the domain of f -1 and consider
whether any of those values should be excluded because of the context. The
inverse represents the number of seconds the button should be pressed to
enlarge the image by a scale factor of x. It is reasonable to say that the scale factor
can be no smaller than 1, so the domain of f -1 is x Ú 1. The domain of f is the
range of f -1 , y Ú 0, which is reasonable as well.
B Sketch the graphs of f and f −1 on a coordinate plane.
Step 1 Graph f(x) = 10x for x Ú 0.
1100
8
y
Step 2 Reflecting across
the line y = x produces
the inverse of f(x) = 10x.
(1, 10)
ff(x)
f(
(x) = 10
1 x
6
y=x
4
2
(0, 1)
f−1(x) = log x
((10,
10, 1))
10,
x
2
(1, 0)
4
6
8
100
Step 3 Choose a few points on
f(x) = 10x and reverse their
coordinates. Plot these
new points and graph
f-1(x) = log x.
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Problem 4
P
TEKS Process Standard (1)(D)
Analyzing Attributes of Logarithmic Functions
Sketch the graphs of f (x) = log 2 x and g(x) = log 10 x each on a separate
coordinate plane. Use the graphs to answer the following questions.
A Identify any asymptotes and intercepts of the graphs. Are any of these
characteristics the same for both functions?
f (x)
How can you use the
graphs of y = 2x
and y = 10x to
generate the graphs
of y = log2 x and
y = log10 x?
The logarithmic functions
are the inverses of y = 2x
and y = 10x , so their
graphs will be reflections
of the exponential
function graphs across
the line y = x.
4
g(x)
y
4
2
2
O
y
2
4
6
8
x
10
⫺2
O
2
4
6
x
10
8
-2
From the graphs, it is clear that x = 0 is a vertical asymptote of both graphs.
Neither function has a y-intercept.
The x-intercept of both graphs is (1, 0).
B State the domain and range of each function.
For f (x), the domain is x 7 0, and the range is the set of all real numbers.
For g(x), the domain is x 7 0, and the range is the set of all real numbers.
Problem
bl
5
Finding the Maximum and Minimum of Logarithmic Functions
Consider the functions f (x) = log 2 x and g(x) = log 10 x on the interval (0, 12].
A Graph the functions on a graphing calculator. Use the same appropriate
window for both functions.
How can you enter
log2 x on your
graphing calculator?
Some calculators have
a log b feature. Look for
the feature listed under
MATH.
WINDOW
Xmin = 0
Xmax = 12
Xscl = 1
Ymin = –2
Ymax = 4
Yscl = 1
Xres = 1
Choose a window. Set the y-values
according to the ranges of the functions
for the domain 0 6 x … 12.
Graph the functions. The graph of
f (x) is above the graph of g(x) for
x 7 1.
continued on next page ▶
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Lesson 7-5
Attributes of Logarithmic Functions
Problem 5
continued
B Find the maximum and minimum values of the functions on the interval [4, 10].
If necessary, round to the nearest thousandth.
Y2=log(X)
X =4
Y1=logBASE(X, 2)
X=10
Y=.60205999
When x = 4, log 2 4 = 2.
When x = 4, log 10 4≈0.602.
When x = 10, log 2 10≈3.322.
When x = 10, log 10 10 = 1.
The minimum value of f (x)
on the interval [4, 10] is 2.
The maximum value of f (x)
on the interval [4, 10] is
approximately 3.322.
The minimum value of g (x)
on the interval [4, 10] is
approximately 0.602.
The maximum value of g (x)
on the interval [4, 10] is 1.
NLINE
HO
ME
RK
O
Y=3.3219281
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
You can write 53 = 125 in logarithmic form using the fact that log b b x = x.
log 5 (53) = log 5 (125)
For additional support when
completing your homework,
go to PearsonTEXAS.com.
3 = log 5 125
Apply the log base 5 to each side.
Use log b bx = x to simplify.
Use this method to write each equation in logarithmic form. Show your work.
1. 34 = 81
2. x4 = y
Evaluate each logarithm by rewriting it as a corresponding exponential expression.
3. log 2 16
4. log 4 2
5. log 5 ( -25)
6. log 3 9
7. Identify the type of relationship represented by the graph.
Then describe the inverse of that function. Write the equation
of f -1(x), describe the relationship it represents, state its
domain and range, and sketch its graph.
10
y
(1, 10)
8
6
4
2
O
(0, 1)
2
4
6
8
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10
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8. Analyze Mathematical Relationships (1)(F) Describe and analyze why the
graph of f (x) = log x has an x-intercept but the graph of g(x) = 10x does not.
9. Analyze Mathematical Relationships (1)(F) Describe and analyze why the
graph of g(x) = 10x has a y-intercept but the graph of f (x) = log x does not.
10. The area affected by an oil spill increases by
a factor of 10 every day. Write a function f (x)
to model the area affected after x days.
What type of relationship exists between
the number of days passed and the area
of the affected region?
Day 2
Day 1
Write each equation in logarithmic form.
1
14. 10 = 10-1
For each function and domain, sketch the graph and state the range of the function.
11. 49 = 72
12. 103 = 1000
13. 625 = 54
15. f (x) = log 2 x - 3; domain: [0.5, 8]
16. f (x) = log 10 x; domain: [10, 100]
Consider the following transformations of the function f (x) = log 2 x.
Determine the domain and range of each function shown, then describe
any asymptotes and intercepts.
17.
4
2
(0, 0)
O
18.
y
(7, 3)
(1, 1)
2
4
y
O
x
6
2
−2
8
⫺2
−4
2
(8, 0) x
6
8
4
(2, −2)
(1, −3)
Find the inverse of each function.
19. y = log 4 x
20. y = log 0.5 x
21. y = log 10 x
22. y = log 2 2x
23. Match each function with the graph of its inverse.
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a. y = log 3 x
b. y = log 2 4x
I.
II.
Lesson 7-5 Attributes of Logarithmic Functions
c. y = log 1 x
III.
2
Write each equation in exponential form.
24. log 2 128 = 7
26. log 6 6 = 1
25. log 0.0001 = -4
27. log 4 1 = 0
1
29. log 2 2 = -1
28. log 7 16,807 = 5
30. log 3 19 = -2
31. log 10 = 1
Find the minimum and maximum value of f (x) = log 2 x and g (x) = log 10 x
on the given domain. If necessary, round to the nearest thousandth.
32. [1000, 2000]
33. [0.5, 12]
34. [ 110, 64]
35. Is it possible to find an interval on which f (x) = log 2 x and g(x) = log 10 x have
the following? Explain your reasoning.
r The same minimum value
r The same maximum value
r The same minimum and maximum value
TEXAS Test Practice
T
36. Which is the logarithmic form of the exponential equation 23 = 8?
A. log 8 2 = 3
B. log 8 3 = 2
C. log 3 8 = 2
D. log 2 8 = 3
37. Dan will begin advertising his video production business online using a pay-perclick method, which charges $30 as an initial fee, plus a fixed amount each time
the ad is clicked. Dan estimates that with the cost of 8 cents per click, his ad will
be clicked about 150 times per day. Which expression represents Dan’s total
estimated cost of advertising, in dollars, after x days?
F. (30 + 0.08x)150
G. 360x
H. 30 + 1200x
J. 30 + 12x
38. Which translation takes y = 0 x 0 to y = 0 x + 3 0 - 1?
A. 3 units right, 1 unit down
C. 3 units left, 1 unit down
B. 3 units right, 1 unit up
D. 3 units left, 1 unit up
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