Lesson 21: The Graph of the Natural Logarithm Function
Lesson 21
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
Lesson 21: The Graph of the Natural Logarithm Function Classwork Exploratory Challenge Your task is to compare graphs of base ππ logarithm functions to the graph of the common logarithm function ππ(π₯π₯) = log(π₯π₯) and summarize your results with your group. Recall that the base of the common logarithm function is 10. A graph of ππ is provided below. a.
Select at least one base value from this list:
1
10
,
ππ(π₯π₯) = log ππ (π₯π₯) for your selected base value, ππ.
b.
1 2
, 2, 5, 20, 100. Write a function in the form
Graph the functions ππ and ππ in the same viewing window using a graphing calculator or other graphing application, and then add a sketch of the graph of ππ to the graph of ππ shown below. y
4 2
4
2
6
8
10
12
14
16
18
20
x
-2 -4
c.
Describe how the graph of ππ for the base you selected compares to the graph of ππ(π₯π₯) = log(π₯π₯).
Lesson 21: Date:
The Graph of the Natural Logarithm Function 10/29/14
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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
S.139
Lesson 21
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
d.
Share your results with your group and record observations on the graphic organizer below. Prepare a group presentation that summarizes the groupβs findings. How does the graph of ππ(ππ) = π₯π₯π₯π₯π₯π₯ ππ (ππ) compare to the graph of ππ(ππ) = π₯π₯π₯π₯π₯π₯(ππ) for various values of ππ? 0 < ππ < 1
1 < ππ < 10
ππ > 10
Exercise 1 Use the change of base property to rewrite each function as a common logarithm. Base ππ
Base 10 (Common Logarithm)
U
ππ(π₯π₯) = log 1 (π₯π₯) 4
ππ(π₯π₯) = log 1 (π₯π₯) 2
ππ(π₯π₯) = log 2 (π₯π₯) ππ(π₯π₯) = log 5 (π₯π₯) ππ(π₯π₯) = log 20 (π₯π₯) ππ(π₯π₯) = log100 (π₯π₯) Lesson 21: Date:
The Graph of the Natural Logarithm Function 10/29/14
Β© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
S.140
Lesson 21
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
Example 1: The Graph of the Natural Logarithm Function ππ(ππ) = π₯π₯π₯π₯(ππ)
Graph the natural logarithm function below to demonstrate where it sits in relation to the base 2 and base 10 logarithm functions. y 4 2
4
2
6
8
10
12
14
16
18
20
x
-2 -4
Example 2 Graph each function by applying transformations of the graphs of the natural logarithm function. a.
ππ(π₯π₯) = 3 ln(π₯π₯ β 1)
y 4
2
-2
2
4
6
8
10
x
-2
-4
Lesson 21: Date:
The Graph of the Natural Logarithm Function 10/29/14
Β© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
S.141
Lesson 21
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
b.
ππ(π₯π₯) = log 6 (π₯π₯) β 2
y 4
2
-2
2
4
6
8
10
x
-2
-4
Lesson 21: Date:
The Graph of the Natural Logarithm Function 10/29/14
Β© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
S.142
Lesson 21
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
Problem Set 1.
Rewrite each logarithm function as a natural logarithm function. a. b. c. d. e. f.
2.
ππ(π₯π₯) = log 5 (π₯π₯)
ππ(π₯π₯) = log 2 (π₯π₯ β 3) π₯π₯
ππ(π₯π₯) = log 2 οΏ½ οΏ½ 3
ππ(π₯π₯) = 3 β log(π₯π₯)
ππ(π₯π₯) = 2log(π₯π₯ + 3) ππ(π₯π₯) = log 5 (25π₯π₯)
Describe each function as a transformation of the natural logarithm function ππ(π₯π₯) = ln(π₯π₯). a.
b. c.
d.
ππ(π₯π₯) = 3 ln(π₯π₯ + 2) ππ(π₯π₯) = βln(1 β π₯π₯)
ππ(π₯π₯) = 2 + ln(ππ 2 π₯π₯)
ππ(π₯π₯) = log 5 (25π₯π₯)
3.
Sketch the graphs of each function in Problem 2 and identify the key features including intercepts, decreasing or increasing intervals, and the vertical asymptote.
4.
Solve the equation ππ βπ₯π₯ = ln(π₯π₯) graphically.
5. 6.
Use a graphical approach to explain why the equation log(π₯π₯) = ln(π₯π₯) has only one solution.
Juliet tried to solve this equation as shown below using the change of base property and concluded there is no solution because ln(10) β 1. Construct an argument to support or refute her reasoning. log(π₯π₯) = ln(π₯π₯)
οΏ½
7.
ln(π₯π₯) = ln(π₯π₯) ln(10)
1 1 ln(π₯π₯) οΏ½ = (ln(π₯π₯)) ln(π₯π₯) ln(10) ln(π₯π₯) 1 =1 ln(10)
Consider the function ππ given by ππ(π₯π₯) = log π₯π₯ (100) for π₯π₯ > 0 and π₯π₯ β 1. a.
b. c. d. e.
What are the values of ππ(100), ππ(10), and πποΏ½β10οΏ½?
Why is the value 1 excluded from the domain of this function? Find a value π₯π₯ so that ππ(π₯π₯) = 0.5.
Find a value π€π€ so that ππ(π€π€) = β1.
Sketch a graph of π¦π¦ = log π₯π₯ (100) for π₯π₯ > 0 and π₯π₯ β 1. Lesson 21: Date:
The Graph of the Natural Logarithm Function 10/29/14
Β© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
S.143
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
Lesson 21: The Graph of the Natural Logarithm Function Classwork Exploratory Challenge Your task is to compare graphs of base ππ logarithm functions to the graph of the common logarithm function ππ(π₯π₯) = log(π₯π₯) and summarize your results with your group. Recall that the base of the common logarithm function is 10. A graph of ππ is provided below. a.
Select at least one base value from this list:
1
10
,
ππ(π₯π₯) = log ππ (π₯π₯) for your selected base value, ππ.
b.
1 2
, 2, 5, 20, 100. Write a function in the form
Graph the functions ππ and ππ in the same viewing window using a graphing calculator or other graphing application, and then add a sketch of the graph of ππ to the graph of ππ shown below. y
4 2
4
2
6
8
10
12
14
16
18
20
x
-2 -4
c.
Describe how the graph of ππ for the base you selected compares to the graph of ππ(π₯π₯) = log(π₯π₯).
Lesson 21: Date:
The Graph of the Natural Logarithm Function 10/29/14
Β© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
S.139
Lesson 21
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
d.
Share your results with your group and record observations on the graphic organizer below. Prepare a group presentation that summarizes the groupβs findings. How does the graph of ππ(ππ) = π₯π₯π₯π₯π₯π₯ ππ (ππ) compare to the graph of ππ(ππ) = π₯π₯π₯π₯π₯π₯(ππ) for various values of ππ? 0 < ππ < 1
1 < ππ < 10
ππ > 10
Exercise 1 Use the change of base property to rewrite each function as a common logarithm. Base ππ
Base 10 (Common Logarithm)
U
ππ(π₯π₯) = log 1 (π₯π₯) 4
ππ(π₯π₯) = log 1 (π₯π₯) 2
ππ(π₯π₯) = log 2 (π₯π₯) ππ(π₯π₯) = log 5 (π₯π₯) ππ(π₯π₯) = log 20 (π₯π₯) ππ(π₯π₯) = log100 (π₯π₯) Lesson 21: Date:
The Graph of the Natural Logarithm Function 10/29/14
Β© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
S.140
Lesson 21
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
Example 1: The Graph of the Natural Logarithm Function ππ(ππ) = π₯π₯π₯π₯(ππ)
Graph the natural logarithm function below to demonstrate where it sits in relation to the base 2 and base 10 logarithm functions. y 4 2
4
2
6
8
10
12
14
16
18
20
x
-2 -4
Example 2 Graph each function by applying transformations of the graphs of the natural logarithm function. a.
ππ(π₯π₯) = 3 ln(π₯π₯ β 1)
y 4
2
-2
2
4
6
8
10
x
-2
-4
Lesson 21: Date:
The Graph of the Natural Logarithm Function 10/29/14
Β© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
S.141
Lesson 21
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
b.
ππ(π₯π₯) = log 6 (π₯π₯) β 2
y 4
2
-2
2
4
6
8
10
x
-2
-4
Lesson 21: Date:
The Graph of the Natural Logarithm Function 10/29/14
Β© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
S.142
Lesson 21
NYS COMMON CORE MATHEMATICS CURRICULUM
M3
ALGEBRA II
Problem Set 1.
Rewrite each logarithm function as a natural logarithm function. a. b. c. d. e. f.
2.
ππ(π₯π₯) = log 5 (π₯π₯)
ππ(π₯π₯) = log 2 (π₯π₯ β 3) π₯π₯
ππ(π₯π₯) = log 2 οΏ½ οΏ½ 3
ππ(π₯π₯) = 3 β log(π₯π₯)
ππ(π₯π₯) = 2log(π₯π₯ + 3) ππ(π₯π₯) = log 5 (25π₯π₯)
Describe each function as a transformation of the natural logarithm function ππ(π₯π₯) = ln(π₯π₯). a.
b. c.
d.
ππ(π₯π₯) = 3 ln(π₯π₯ + 2) ππ(π₯π₯) = βln(1 β π₯π₯)
ππ(π₯π₯) = 2 + ln(ππ 2 π₯π₯)
ππ(π₯π₯) = log 5 (25π₯π₯)
3.
Sketch the graphs of each function in Problem 2 and identify the key features including intercepts, decreasing or increasing intervals, and the vertical asymptote.
4.
Solve the equation ππ βπ₯π₯ = ln(π₯π₯) graphically.
5. 6.
Use a graphical approach to explain why the equation log(π₯π₯) = ln(π₯π₯) has only one solution.
Juliet tried to solve this equation as shown below using the change of base property and concluded there is no solution because ln(10) β 1. Construct an argument to support or refute her reasoning. log(π₯π₯) = ln(π₯π₯)
οΏ½
7.
ln(π₯π₯) = ln(π₯π₯) ln(10)
1 1 ln(π₯π₯) οΏ½ = (ln(π₯π₯)) ln(π₯π₯) ln(10) ln(π₯π₯) 1 =1 ln(10)
Consider the function ππ given by ππ(π₯π₯) = log π₯π₯ (100) for π₯π₯ > 0 and π₯π₯ β 1. a.
b. c. d. e.
What are the values of ππ(100), ππ(10), and πποΏ½β10οΏ½?
Why is the value 1 excluded from the domain of this function? Find a value π₯π₯ so that ππ(π₯π₯) = 0.5.
Find a value π€π€ so that ππ(π€π€) = β1.
Sketch a graph of π¦π¦ = log π₯π₯ (100) for π₯π₯ > 0 and π₯π₯ β 1. Lesson 21: Date:
The Graph of the Natural Logarithm Function 10/29/14
Β© 2014 Common Core, Inc. Some rights reserved. commoncore.org
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
S.143
