7-8
Attributes and Transformations of the
Natural Logarithm Function
TEKS FOCUS
VOCABULARY
ĚNatural logarithm function – A natural logarithm
TEKS (5)(A) Determine the effects on the key attributes on the
graphs of f(x) = bx and f(x) = logb (x) where b is 2, 10, and e when
f(x) is replaced by af(x), f(x) + d, and f(x - c) for specific positive
and negative real values of a, c, and d.
function is a logarithm function with base e. The
natural logarithm function, y = ln x, is y = loge x.
It is the inverse of y = ex .
TEKS (1)(E) Create and use representations to organize, record,
and communicate mathematical ideas.
ĚRepresentation – a way to display or describe
information. You can use a representation to
present mathematical ideas and data.
Additional TEKS (1)(D), (2)(A)
ESSENTIAL UNDERSTANDING
Logarithm functions with base e have the same properties as other logarithm
functions. The functions y = ex and y = log e x are inverse functions. The relationship
a = eb can be rewritten as b = log e a.
Key Concept
Natural Logarithm Function
If y = ex , then x = log e y = ln y. The natural logarithm function is the inverse of
y = e x . You can write the natural logarithm function as y = log e x or y = ln x.
f(x) = e x
ex
x
312
Lesson 7-8
21
1
e
0
1
1
e
≈1.39
4
2
e2
f−1(x) = ln x
Graph
4
2
(0, 1)
-4
-2
O
y
y = ex
(1.39, 4)
(4, 1.39)
x
loge x
1
e
-1
x
1
0
e
1
4
≈1.39
e2
2
2
4
(1, 0)
-2
y = loge x
-4
Attributes and Transformations of the Natural Logarithm Function
Key Concept
Attributes of the Natural Logarithm Function
The graph of f (x) = log e x has these attributes.
r Domain: (0, ∞)
r Range: all real numbers
r x-intercept: (1, 0)
r y–intercept: none
r asymptote: x = 0
4
y
x-intercept
2
x
-44 -22 O
2
4
asymptote
-22 y = loge x
x=0
-44
Problem 1
P
TEKS Process Standard (1)(D)
Domain, Range, Intercepts, and Asymptotes of y = ln x
Graph the function f (x) = ln x and analyze the domain, range, intercepts,
and asymptote.
Step 1 Make a table. The equation f (x) = ln x
can also be written as the
function f (x) = log e x.
x
How can you use
the graph to help
you determine
the domain of the
function?
The graph shows that line
x = 0 is an asymptote.
All values of x for the
function are greater
than zero.
Step 2 Then sketch the graph.
loge x
2
-2
O
21 undefined
-2
0 undefined
-4
1
e
21
-6
1
0
e
1
e2
2
y
2
4
6
8
x
10
Step 3 Analyze the function.
S
domain: x 7 0
Where f(x) is defined
range: all real numbers
The set of values of f(x)
x-intercept: (1, 0)
Where f(x) crosses the x-axis
asymptote: x = 0
The line that f(x) approaches, but does not cross
The
T graph of f (x) = ln x shows that the domain is x 7 0, the range is all real
numbers, the x-intercept is at (1, 0), and x = 0 is a vertical asymptote.
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Problem 2
P
Maximum and Minimum for a Given Interval of f (x) = loge x
For the interval
[1, 2], is loge 1 the
minimum and loge 2
the maximum of
f(x) = loge x?
Yes, since the function
loge x always increases
for x 7 0. So 0 6 1 6 2
means f(1) 6 f(2).
Find the maximum and minimum of log e x over the interval [1, 2].
F
The
T maximum is at the end of the interval, or approximately 0.693.
2
y
The
T minimum is at the beginning of the interval, or 0.
1
x
-1
O
1
2
-1
-2
Problem
bl
3
TEKS Process Standard (1)(E)
Analyzing y = af (x) for f (x) = loge x
What do the graphs
for y = a loge x and
y = loge x have in
common?
Both graphs pass through
(1, 0) and have x = 0 as
an asymptote.
Graph the parent function y = log e x, and then graph y = a ~ log e x for different
G
values
of a on the same set of axes. What is the effect of the transformation on the
v
x-intercept?
x
A y = 0.5 log e x
Multiplying the parent function by 0.5
will compress the graph of the function
vertically because 0 a 0 6 1.
y
y 5 loge x
1
x
O
Start at (1, 0). Plot all points of log e x at 12
their value.
⫺1
The x-intercept remains at (1, 0).
⫺2
1
2
3
4
y 5 −2 loge x
B y = −2 log e x
Multiplying the parent function by -2 will stretch the graph of the function
vertically because 0 a 0 7 1. It will also reflect the graph of the parent function
across the x-axis because a 6 0.
Start at (1, 0). Plot all points of log e x at twice their value and on the opposite
side of the x-axis.
The x-intercept remains at (1, 0).
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Lesson 7-8
Attributes and Transformations of the Natural Logarithm Function
y 5 0.5 loge x
Problem 4
P
TEKS Process Standard (1)(E)
Analyzing y = f (x) + d for f (x) = loge x
Do the graphs of
y = loge x and
y = loge x + d ever
intersect?
No. Although they
approach the same
asymptote, they do not
pass through the same
points.
Graph the parent function y = log e x, and then graph y = log e x + d for different
values of d on the same set of axes. What is the effect of the transformation on the
domain?
d
A y = log e x + 3
4
Because d 7 0, the graph translates up from
the parent function.
y
y 5 loge x + 3
2
y 5 loge x
x
Plot the point that is 3 up from (1, 0), which is
the point (1, 3). Plot the other points of log e x
by moving up by 3 units.
⫺2
The domain remains x 7 0.
⫺4
O
2
4
6
8
y 5 loge x − 2
B y = log e x − 2
Because d 6 0, the graph translates down from the parent function.
Plot the point that is 2 down from (1, 0), which is the point (1, -2).
Plot the other points of log e x by moving down by 2 units.
The domain remains x 7 0.
Problem
bl
5
TEKS Process Standard (1)(E)
Analyzing y = f (x − c) for f (x) = loge x
How do you know
the graph of
y = loge (x − 3) is
correct?
Since c = 3 and c 7 0,
the graph of the parent
function will shift 3 units
to the right.
Graph the parent function y = log e x, and then graph y = log e (x − c) for different
values of c on the same set of axes. What is the effect of the transformation on the
asymptote?
a
A y = log e (x − 3)
4
Because c 7 0, the translation is to the right.
Start at (1, 0), and move 3 units to the right
to plot (4, 0). Plot all points of log e x to the
right by 3 units.
The asymptote shifts 3 units to the right, to
x = 3.
2
y
y 5 loge (x + 2)
y 5 loge x
x
⫺2
O
⫺2
2
4
6
8
y 5 loge (x − 3)
⫺4
B y = log e (x + 2)
Because c 6 0, the translation is to the left.
Start at (1, 0), and move 2 units to the left to plot ( -1, 0). Plot all points of log e x
to the left by 2 units.
The asymptote shifts 2 units to the left, to x = -2.
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Problem 6
P
Using Transformations of Natural Logarithm Functions
The time T in years at which an investment is worth x dollars is given by the
function T = 20 ln x − 160. Describe the graph of this function as a translation,
stretch, compression, or reflection of the parent function, T = ln x.
The transformation of the log function is T = a ln x + d. In this example, a = 20, which
indicates a vertical stretch; and d = -160, which indicates a translation down 160 units.
NLINE
HO
ME
RK
O
The combined transformation of the parent function T = ln x is a vertical stretch
by a factor of 20 and a translation of 160 units (years) down.
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Analyze the domain, range, x-intercept, and asymptote of the graphs of
the function.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
1
1. f (x) = log e x
2. f (x) = - 8 ln x
4. f (x) = log e x - 1
3. f (x) = 1.5 ln (x + 5)
5. f (x) = -4 log e (x - 1.5)
1
6. f (x) = 2 log e x + 1
Create Representations to Communicate Mathematical Ideas (1)(E) Sketch
the graphs of each pair of functions on the same coordinate grid. Identify the
transformation that maps the first function to the second function. Determine
the effect the transformation has on the intercepts.
7. f (x) = log e x, g(x) = 2 log e x
1
8. f (x) = log e x, g(x) = - 3 log e x
9. f (x) = log e x, g(x) = log e x - 13
10. f (x) = log e x, g(x) = log e (x + 4)
11. f (x) = -log e x, g(x) = -log e (x - 3)
12. f (x) = 2 log e x, g(x) = -2 log e x - 5
13. f (x) = log e x, g(x) = -0.5 log e x + 0.5
14. Analyze Mathematical Relationships (1)(F) A transformation of the graph
of f (x) = log e x passes through the point (1, 8). Does the function f (x) + 8
describe the only possible transformation of the graph of f (x)? Explain.
Let f (x) = log e x. Identify the minimum and maximum values over the specified
interval, or state that a maximum or minimum does not exist.
316
Lesson 7-8
15. 1 … x … e2
16. e2 … x … e3
17. 5 … x … 6
18. 0.001 … x … 0.002
19. 0 … x … 1
20. -1 … x … 0
Attributes and Transformations of the Natural Logarithm Function
21. Explain Mathematical Ideas (1)(G) A transformation of the graph of f (x) = log e x
passes through the point (1, 0). Is the transformation described by a function of
the form g(x) = a log e x, h(x) = a log e (x - c), or j(x) = a log e x + d? Assume that
a, c, and d are not equal to zero. Explain your answer.
TEXAS Test Practice
T
22. The graph of f (x) = log e x and a graph of a transformation of f (x) are shown.
What is the transformed function?
4
y
2
x
O
2
4
6
⫺2
⫺4
A. f (x) = -0.5 log e x
C. f (x) = log e x + 4
B. f (x) = -5 log e x
D. f (x) = 2 log e (x - 4)
23. Which transformation maps the graph of f (x) = log e x to the graph of
g(x) = log e (x + 13)?
F. a translation of 13 units to the left
G. a translation of 13 units down
H. a compression by a factor of 13
J. a translation of 13 units to the right
24. What is the domain of the function f (x) = -3 log e (x + 5) - 8?
A. x 7 -8
C. x 7 0
B. x 7 -5
D. x 7 3
25. Let n be a nonzero real number. Of the functions y = log e x + n; y = log e (x - n);
and y = n log e x, which has a different asymptote from the parent function
y = log e x? Explain.
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