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General Mathematics
Quarter 1- Module 9: Graph of Logarithmic Function
Senior High School
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Core Subject SHS- General Math (Grade 11) Alternative Delivery
Mode Quarter 1 - Module 9: representing a logarithmic function
through its: (a) table of values, (b) graph, and (c) equation,
finding the domain and range of a logarithmic function and
determining the intercepts, zeroes and asymptotes of a logarithmic
function First Edition, 2020 Republic Act 8293, section 176 states
that: No copyright shall subsist in any work of the Government of
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to locate and seek permission to use these materials from their
respective copyright owners. The publisher and authors do not
represent nor claim ownership over them. Published by the
Department of Education Secretary: Leonor Magtolis Briones
Undersecretary: Diosdado M. San Antonio Printed in the Philippines
by: Department of Education – Region VII, Division of Cebu Province
Office Address: IPHO Bldg., Sudlon, Lahug, Cebu City, Philippines
Telefax: (032) 255 - 6405 E-mail Address:
[email protected]
Development Team of the Module Writer: Ma. Bukcemas D. Cahutay
Reviewer: Dr. Clavel D. Salinas Illustrator: Ma. Bukcemas D.
Cahutay Layout Artist: Ma. Bukcemas D. Cahutay Evaluator: Henry D.
Espina Jr. Moderator: Dr. Arlene D. Buot
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Chief, CID: Dr. Mary Ann P. Flores EPS in LRMS: Mr. Isaiash T.
Wagas PSDS/ SHS Division Coordinator: Dr. Clavel D. Salinas
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General Mathematics
Quarter 1- Module 9: Graph of Logarithmic Function
Senior High School
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Introductory Message
For Educators:
Learning is a constant process. Amidst inevitable circumstances,
Department of
Education extends their resources and looks for varied ways to
cater your needs and to adapt
to the new system of Education as a fortress of Learning
Continuity Plan. One of the probable
solutions is the use of Teacher-made Educational Modules in
teaching.
You are reading the General Mathematics- Grade 11: First Quarter
Alternative
Delivery Mode (ADM) Module on representing a logarithmic
function through its: (a) table
of values, (b) graph, and (c) equation (M11GM-Ii-2), finding the
domain and range of a
logarithmic function (M11GM-Ii-3) and determining the
intercepts, zeroes and asymptotes of
an logarithmic function (M11GM-Ii-4), as written and found in
the K-12 Most Essential
Learning Competencies.
The creation of this module is a combined effort of competent
educators from
different levels and various schools of Department of
Education-Cebu Province. In addition,
this module is meticulously planned, organized, checked and
verified by knowledgeable
educators to assist you in imparting the lessons to the learners
while considering the physical,
social and economical restraints in teaching process.
The use of Teacher-made Educational Module aims to surpass the
challenges of
teaching in a new normal education set-up. Through this module,
the students are given
independent learning activities, which embodies in the Most
Essential Learning
Competencies based from the K-12 Curriculum Competencies, to
work on in accordance to
their capability, efficiency and time. Thus, helping the
learners acquire the prerequisite 21st
Century skills needed with emphasis on utmost effort in
considering the whole well being of
the learners.
As the main source of learning, it is your top priority to
explain clearly on how to use
this module to the learners. While using this module, learner’s
progress and development
should be recorded verbatim to assess their strengths and
weaknesses while doing the
activities presented independently in safety of their homes.
Moreover, you are anticipated to
persuade learners to comply and to finish the modules on or
before the scheduled time.
For the Learners:
As a significant stakeholder of learning, Department of
Education researched and
explored on innovative ways to address your needs with high
consideration on social,
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economic, physical and emotional aspects of your well being. To
continue the learning
process, DepEd comes up with an Alternative Delivery mode of
teaching using Teacher-
Made Educational Modules.
You are reading the General Mathematics- Grade 11: First Quarter
Alternative
Delivery Mode (ADM) Module representing a logarithmic function
through its: (a) table of
values, (b) graph, and (c) equation (M11GM-Ii-2), finding the
domain and range of a
logarithmic function (M11GM-Ii-3) and determining the
intercepts, zeroes and asymptotes of
an logarithmic function (M11GM-Ii-4), as written and found in
the K-12 Most Essential
Learning Competencies.
This module is especially crafted for you to grasp the
opportunity to continue learning
even at home. Using guided and independent learning activities,
rest assured that you will be
able to take pleasure as well as to deeply understand the
contents of the lesson presented;
recognizing your own capacity and capability in acquiring
knowledge.
This module has the following parts and corresponding icons:
The first part of the module will keep you on track
on the Competencies, Objectives and Skills
expected for you to be developed and mastered.
This part aims to check your prior knowledge on
the lesson to take.
This part helps you link the previous lesson to the
current one through a short exercise/drill.
The lesson to be partaken is introduced in this part
of the module creatively. It may be through a story,
a song, a poem, a problem opener, an activity, a
situation or the like.
A brief discussion of the lesson can be read in this
part. It guides and helps you unlock the lesson
presented.
WHAT I NEED TO KNOW
WHAT I KNOW
WHAT’S IN
WHAT’S NEW
WHAT IS IT
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A comprehensive activitiy/es for independent
practice is in this part to solidify your knowledge
and skills of the given topic.
This part of the module is used to process your
learning and understanding on the given topic.
A transfer of newly acquired knowledge and skills
to a real-life situation is present in this part of the
module.
This activity assesses your level of mastery
towards the topic.
In this section, enhancement activities will be
given for you to further grasp the lessons.
This contains answers to all activities in the
module.
At the end of this module you will also find:
References Printed in this part is a list of all reliable and
valid
resources used in crafting and designing this
module.
WHAT’S MORE
WHAT I HAVE LEARNED
WHAT I CAN DO
ASSESSMENT
ADDITIONAL ACTIVITIES
ANSWER KEY
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In using this module, keep note of the fundamental reminders
below.
1. The module is government owned. Handle it with care.
Unnecessary
marks are prohibited. Use a separate sheet of paper or notebook
in
answering all the given exercises.
2. This module is organized according to the level of
understanding.
Skipping one part of this module may lead you to confusion
and
misinterpretation.
3. The instructions are carefully laden for you to understand
the given
lessons. Read each item cautiously.
4. This is a Home-Based class, your reliability and honour in
doing the
tasks and checking your answers are a must.
5. This module helps you attain and learn lessons at home. Make
sure to
clearly comprehend the first activity before proceeding to the
next one.
6. This module should be returned in good condition to your
teacher/facilitator once you completed it.
7. Answers should be written on a separate sheet of paper or
notebook
especially prepared for General Mathematics subject.
If you wish to talk to your teacher/educator, do not hesitate to
keep in
touch with him/her for further discussion. Know that even if
this is a home-
based class, your teacher is only a call away. Good
communication between
the teacher and the student is our priority to flourish your
understanding on
the given lessons.
We do hope that in using this material, you will gain ample
knowledge
and skills for you to be fully equipped and ready to answer the
demands of
the globally competitive world. We are confident in you! Keep
soaring high!
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Good day dear learner!
This module is solely prepared for you to access and to acquire
lessons
befitted in your grade level. The exercises, drills and
assessments are
carefully made to suit your level of understanding. Indeed, this
learning
resource is for you to fully comprehend representing a
logarithmic function
through its: (a) table of values, (b) graph, and (c) equation
(M11GM-Ii-2),
finding the domain and range of a logarithmic function
(M11GM-Ii-3) and
determining the intercepts, zeroes and asymptotes of a
logarithmic function
(M11GM-Ii-4), as written and found in the K-12 Most Essential
Learning
Competencies. Independently, you are going to go through this
module
following its proper sequence. Although you are going to do it
alone, this is a
guided lesson and instructions/directions on how to do every
activity is plotted
for your convenience.
PERFORMANCE STANDARD:
In this module, the learner is able to apply the concepts of
logarithmic functions, and
to formulate and solve real-life problems with precision and
accuracy.
MOST ESSENTIAL LEARNING COMPETENCY:
The learner must be able to represent a logarithmic function
through its: (a) table of
values, (b) graph, and (c) equation (M11GM-Ii-2), find the
domain and range of a
logarithmic function (M11GM-Ii-3) and determine the intercepts,
zeroes and asymptotes of a
logarithmic function (M11GM-Ii-4)
LESSON AND COVERAGE:
LESSON INTENDED LEARNING OUTCOMES
The learner:
Graphing a Logarithmic Function • represents a logarithmic
function through
its (a) table of values, (b) graph, and (c)
equation;
• sketches the graph of logarithmic function
• finds the domain and range of a logarithmic
WHAT I NEED TO KNOW
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function;
• determines the intercepts, zeroes, and
asymptotes of a logarithmic function; and,
• shows perseverance in graphing logarithmic
function
MOST ESSENTIAL LEARNING COMPETENCY:
The learner must be able to represent a logarithmic function
through its: (a) table of
values, (b) graph, and (c) equation (M11GM-Ii-2), find the
domain and range of a
logarithmic function (M11GM-Ii-3) and determine the intercepts,
zeroes and asymptotes of a
logarithmic function (M11GM-Ii-4)
Learning Outcome(s):
After going through this module, you are expected to:
• represent a logarithmic function through its (a) table of
values, (b) graph, and (c)
equation;
• sketch the graph of logarithmic function
• find the domain and range of a logarithmic function;
• determine the intercepts, zeroes, and asymptotes of a
logarithmic function; and,
• show perseverance in graphing logarithmic function
Graphing a Logarithmic Function Lesson 1
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In the previous module, you learned about the properties and
laws of logarithms as
well as solving logarithmic equations and inequalities. This
module will illustrate how
logarithmic functions can be represented through table of values
and graph. But before that,
answer first the pre-assessment below to test your prior
knowledge about the topic.
Pre-Assessment:
Directions: Find out how much you already know about graphing
logarithmic functions.
Write the letter that corresponds to your answer in your
notebook. Take note of the items that
you were not able to answer correctly and find the right answer
as you go through this
module.
1. What is the function as shown in the table below?
x 1/4 1/2 1 2 4 8
y -6 -3 0 3 6 9
a. y= 2log3x b. y=3log2x c. y=log3x + 2 d. y= log3(x+2)
2. Which of the following is NOT true to all logarithmic
functions?
a. It is a one-to-one function.
b. There is no horizontal asymptote.
c. The domain is the set of all positive real numbers.
d. The range is the set of all real numbers.
3. Which of the following functions is the reflection of y = 2x
along the line y = x?
a. y = log2x b. x = log2y c. y = logx + 2 d. y = 2logx
4. Find the domain of the function y= log3x?
a. All negative numbers c. All real numbers
b. All positive numbers d. All integers
5. All of the following are true for both functions y= log3x and
y= log1/3x EXCEPT
a. They have the same domain. c. They are both increasing.
b. They have the same x-intercept. d. They have the same
vertical asymptote.
6. Which of the following functions y= log2x shifted 3 units to
the right?
a. y= log2x – 3 b. y= log2x + 3 c. y= log2(x+3) d. y=
log2(x-3)
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7. What is the domain of a logarithmic function y= logbx, given
that 0< b < 1?
a. x>0 b. x 0 b. x < 0 c. x > 2 d. x < 2
9. What is the range?
a. y>0 b. y>2 c. y
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15. Which of the following statement is true about the function
y= log2(x+1)?
a. The domain of the function is the set of all positive
numbers.
b. Its graph intersects the y-axis at (0,0)
c. The graph is asymptotic to the x-axis.
d. There is no x-intercept.
Test I. Directions: Below are a number of statements in Column
1. Write “Agree” or
“Disagree” in Column 2 and support your position in Column 3.
Write your answers in your
notebook.
Statement (column 1) Agree or
Disagree?
(column 2)
Justification (column 3)
1 The value of xblog is not a valid expression
if x is a negative number.
2 01log =b only if the base is greater than 1
3 nbnb =log for any b, 1b
4 It is impossible for logarithms to be equal to
a negative number.
5 It is not possible to compute 0log x .
Test II. Directions: Use the properties of logarithm to expand
the expression as a sum,
difference or multiple of logarithms. Simplify. Write your
answer in your notebook.
6. logaxy 9.
7. logb 10.
8.
Test III. Directions: Given that log 5 = 0.69897, log 7 =
0.84510, log 4 = 0.60206, log 3 =
0.47712 and log 2 = 0.30103. Find the following: (Write your
answer in your notebook.)
11. log 42 14.
12. log 49 15. log 140
13.
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Logarithmic and exponential functions are said to be inverses of
each other. Let us
examine their differences and connection by doing this
activity.
Recall that to find the inverse of a function, one needs to
interchange x and y and
express y as a function of x. Getting the inverse of the
function f(x) = 2x, f(x) = 2x
y = 2x let y = f(x)
x = 2y interchange x and y
y = log2x express into logarithmic form
f-1(x) = log2x replace y with f-1(x)
Thus, the inverse of f(x) = 2x is f(x) = log2x.
Let us try to graph the two functions and examine their graphs
afterwards by doing the
activity below.
Directions: Sketch the graph of y = 2x and y = log2x by doing
the steps below: (Copy and
answer the activity in your notebook.)
Step 1: Fill in the table of values below
y = 2x y = log2x
Step 2:
Sketch the graph:
y = 2x and y = log2x
WHAT’S NEW
Logarithmic Function
If a > 0, a ≠ 1, and x > 0, then the logarithmic function
f(x) with base a is
f(x) = logax
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Step 3: Describe the domain and range, intercepts of both
function
y = 2x y = log2x
Domain: __________________ Domain: __________________
Range: __________________ Range: __________________
x-intercept: ___________ x-intercept: ___________
y-intercept ___________ y-intercept ___________
horizontal asymptote: _______ horizontal asymptote: _______
vertical asymptote: _______ vertical asymptote: _______
Answer the questions that follow:
1. What have you noticed with the graph in relation to the line
y = x?
___________________________________________________________________________
2. What have you noticed with the table of values?
___________________________________________________________________________
3. What have you noticed with their domain and range, intercepts
and asymptote of both
function?
___________________________________________________________________________
Relationship Between the Graphs of Logarithmic and Exponential
Functions
For all positive real numbers x and b, b ≠ 1, there exists a
real number y such that y =
logbx if and only if x = by.
Since logarithmic and exponential functions are inverses of each
other, the graph of a
logarithmic function is obtained by reflecting the graph of the
exponential function about the
graph of y = x, which is the line of symmetry as shown
below:
In the following examples, the graph is obtained by first
plotting a few points. Results
will be generalized later on.
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GRAPHING LOGARITHMIC FUNCTION OF THE FORM y = logbx
Example 1. Sketch the graph of y = log2x.
Solution:
Step 1: Construct a table of values of ordered pairs for the
given function. A table of values
for y = log2x is as follows:
Step 2: Plot the points found in the table, and connect them
using a smooth curve.
It can be observed that the function is defined only for
positive numbers, that is the
domain is x > 0. The function is strictly increasing, and
attains all real values (range is the
set of all real numbers). As x approaches 0 from the right, the
function decreases without
bound, i.e., the line x = 0 is a vertical asymptote.
Example 2. Sketch the graph of y = log1/2x.
Solution:
Step 1: Construct a table of values of ordered pairs for the
given function. A table of values
for y = log1/2x is as follows:
Step 2: Plot the points found in the table, and connect them
using a smooth curve.
x 1/16 1/8 1/4 1/2 1 2 4 8
y -4 -3 -2 -1 0 1 2 3
x 1/16 1/8 1/4 1/2 1 2 4 8
y 4 3 2 1 0 -1 -2 -3
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It can be observed that the function is defined only for x >
0 (domain is the set of
positive numbers). The function is strictly decreasing, and
attains all real values (range is
the set of all real numbers). As x approaches 0 from the right,
the function increases without
bound, i.e., the line x = 0 is a vertical asymptote.
y = logbx (b > 1) y = logbx (0 < b < 1)
Using the graph of y = logbx from the examples, we can deduce
that:
1. The domain is the set of all positive numbers or {x R | x
> 0}.
2. The range is the set of all real numbers or {y R }
3.The x-intercept is 1. Since the graph does not pass through
y-axis, thus, there is no
y-intercept.
4. The vertical asymptote is the line x = 0 (or the y-axis).
There is no horizontal
asymptote.
5. The value of b determines whether the graph is increasing or
decreasing. If b > 1,
then the graph of the function is increasing. If 0 < b <
1, then the graph is decreasing.
GRAPHING LOGARITHMIC FUNCTION OF THE FORM y = a logbx
Example 3. Sketch the graphs of y = 2log2x. Determine the
domain, range, vertical
asymptote, x-intercept, and zero.
Solution:
The graph of y = 2log2x can be obtained from the graph of y =
log2x by multiplying
each y-coordinate by 2, as the following table of signs
shows.
x 1/16 1/8 1/4 1/2 1 2 4 8
log2x -4 -3 -2 -1 0 1 2 3
y = 2log2x -8 -6 -4 -2 0 2 4 6
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The graph is shown below.
Analysis:
a. Domain: {x | x R, x > 0} or positive numbers b. Range : {y
| y R} or the set of real numbers c. Vertical Asymptote: x = 0
d. x-intercept: 1
e. Zero/es: 1
GRAPHING LOGARITHMIC FUNCTION OF THE FORM y = logbx ±k
Example 4. Sketch the graph of y = log3x – 1
Solution:
Sketch the graph of the basic function y = log3x. Note that the
base 3 > 1. The “–1”
means vertical shift downwards by 1 unit.
Some points on the graph of y = log3x are (1,0), (3,1), and
(9,2). Shift these points 1
unit down to obtain (1, –1), (3,0), and (9,1). Plot these
points.
The graph is shown below.
Analysis:
a. Domain: {x | x R, x > 0} or all positive numbers b. Range
: {y | y R} or the set of real numbers c. Vertical Asymptote: x =
0
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d. x-intercept: 3
e. Zero/es: 3
We can solve for the x-intercept algebraically by letting y be
equal to 0 and solve for
x:
y = log3x – 1
0 = log3x – 1
log3x = 1
x = 31 changing to exponential form
x = 3
Example 5. Sketch the graph of y = log0.25(x + 2).
Solution:
Sketch the graph of the basic function y = log0.25x. Note that
the base 0 < 0.25 < 1.
Rewrite the equation, obtaining y = log0.25[x – (–2)]. The “–2”
means a horizontal shift of 2
units to the left.
Some points on the graph of y = log0.25x are (1,0), (4, –1), and
(0.25,1).
Shift these points 2 units to the left to obtain (–1,0), (2,
–1), and (–1.75,1). Plot these
points.
Graph:
Analysis:
a. Domain: {x | x R , x >–2}
(The expression x+2 should be greater than 0 for log0.25(x+2) to
be defined. Hence, x must
be greater than –2.)
b. Range: {y | y R }
c. Vertical Asymptote: x = –2
d. x-intercept: -1
e. Zero/es: -1
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For each of the following functions, (a) fill in the table of
values below; (b) sketch the
graph and (c) identify the domain, range, vertical asymptote and
x-intercept. Copy and
answer the activity in your notebook.
1. y = log3x
Domain: _______________
Range: _______________
Vertical Asymptote: _________
x-intercept: _______________
2. y = log3(x+1)
Domain: _______________
Range: _______________
Vertical Asymptote: _________
x-intercept: _______________
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3. y = 2 log3x
Domain: _______________
Range: _______________
Vertical Asymptote: _________
x-intercept: _______________
These are the summary of the analysis of graph of logarithmic
function.
Directions: Copy the table and fill in the correct information
in your notebook. The first row
is done for you.
Transformation
of f(x) =
logbx
DOMAIN RANGE X-INTERCEPT
f(x) = logbx NONE x > 0 or all
positive
numbers
y R or all real
numbers
1
f(x) = logb(x±k) Moving to
the left or
right
f(x) = a logbx Stretching
f(x) = logbx±k Moving up
or down
Y-INTERCEPT HORIZONTAL ASYMPTOTE
VERTICAL
ASYMPTOTE
f(x) = logbx none none x = 0
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f(x) = logb(x±k)
f(x) = a logbx
f(x) = logbx±k
Directions: Make one example of logarithmic function of any
form: (a) y =
logbx; (b) y = a logbx; (c) y = logbx ± k; and, (d) y =
logb(x±k). Sketch its
graph and identify the x- intercept/zero, vertical asymptote,
domain and range.
Write your answer in your notebook.
Directions: For each of the following functions, (a) construct
the table of values,
(b) sketch the graph and (c) identify the domain, range,
vertical asymptote and
x-intercept.
1. f(x) = log5(x-3) 2. f(x) = log3x + 2 3. f(x) = 4 log2x 4.
f(x) = log2(x-1) + 3
Each function will be rated according to the rubric on the next
page:
WHAT I CAN DO
ASSESSMENT
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Before calculators were invented, people used a table of
logarithms to compute
certain numbers.
Brainstorm and decide how exponents and logarithms can be used
to approximate the
value of the given using the data below. Explain your answer in
your notebook.
log 2 0.3010 log 3 04771 log 5 0.6990 log 3 0.8451
ADDITIONAL ACTIVITIES
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ANSWER KEY
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References
Most Essential Learning Competencies (MELCS)
SHS General Mathematics LM pages 124-133
SHS General Mathematics TG pages 143-152
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For inquiries or feedback, please write or call:
Department of Education: Region VII, Division of Cebu
Province
Office Address: IPHO Bldg., Sudlon, Lahug, Cebu City,
Philippines Telefax: (032) 255 - 6405
Email Address: [email protected]
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