Gen Math Quarter 1-Module 5: Logarithmic Functions, Exercises of Mathematics

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General

Mathematics

Quarter 1 – Module 9:

Logarithmic Function

General Mathematics – Grade 11 Self-Learning Module (SLM) Module 9: Logarithmic Function First Edition, 2020

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Development Team of the Module

Writers: Live C. Angga, Cryzzhylls Jan P. Balonga, Devina Gracia W. Fernandez

Editors: Hannih Lou T. Bantilan

Reviewers: Corinne Anne M. Formoso, Ronel Esto, Cryzzhlls Jan Balonga

Illustrator:

Layout Artist: Maylene F. Grigana

Cover Art Designer: Ian Caesar E. Frondoza

Management Team: Allan G. Farnazo, CESO IV – Regional Director

Fiel Y. Almendra, CESO V – Assistant Regional Director Gildo G. Mosqueda, CEO VI - Schools Division Superintendent Diosdado F. Ablanido, CPA – Assistant Schools Division Superintendent Gilbert B. Barrera – Chief, CLMD Arturo D. Tingson Jr. – REPS, LRMS Peter Van C. Ang-ug – REPS, ADM Jade T. Palomar – REPS, Mathematics Donna S. Panes – Chief, CID Elizabeth G. Torres – EPS, LRMS Judith B. Alba – EPS, ADM Reynaldo C. Tagala – EPS, Mathematics

Introductory Message

For the facilitator:

Welcome to the General Mathematics (Grade 11) Self-Learning Module (SLM) on Logarithmic Functions!

This module was collaboratively designed, developed and reviewed by educators both from public and private institutions to assist you, the teacher or facilitator in helping the learners meet the standards set by the K to 12 Curriculum while overcoming their personal, social, and economic constraints in schooling.

This learning resource hopes to engage the learners into guided and independent learning activities at their own pace and time. Furthermore, this also aims to help learners acquire the needed 21st century skills while taking into consideration their needs and circumstances.

In addition to the material in the main text, you will also see this box in the body of the module:

As a facilitator you are expected to orient the learners on how to use this module. You also need to keep track of the learners' progress while allowing them to manage their own learning. Furthermore, you are expected to encourage and assist the learners as they do the tasks included in the module.

Notes to the Teacher This contains helpful tips or strategies that will help you in guiding the learners.

For the learner:

Welcome to the General Mathematics (Grade 11) Self-Learning Module (SLM) on Logarithmic Functions!

The hand is one of the most symbolized part of the human body. It is often used to depict skill, action and purpose. Through our hands we may learn, create and accomplish. Hence, the hand in this learning resource signifies that you as a learner is capable and empowered to successfully achieve the relevant competencies and skills at your own pace and time. Your academic success lies in your own hands!

This module was designed to provide you with fun and meaningful opportunities for guided and independent learning at your own pace and time. You will be enabled to process the contents of the learning resource while being an active learner.

This module has the following parts and corresponding icons:

What I Need to Know This will give you an idea of the skills or competencies you are expected to learn in the module.

What I Know This part includes an activity that aims to check what you already know about the lesson to take. If you get all the answers correct (100%), you may decide to skip this module.

What’s In This is a brief drill or review to help you link the current lesson with the previous one.

What’s New In^ this^ portion,^ the^ new^ lesson^ will^ be introduced to you in various ways such as a story, a song, a poem, a problem opener, an activity or a situation.

What is It This section provides a brief discussion of the lesson. This aims to help you discover and understand new concepts and skills.

What’s More This^ comprises^ activities^ for^ independent practice to solidify your understanding and skills of the topic. You may check the answers to the exercises using the Answer Key at the end of the module.

What I Have Learned This^ includes^ questions^ or^ blank sentence/paragraph to be filled in to process what you learned from the lesson.

What I Can Do This section provides an activity which will help you transfer your new knowledge or

What I Need to Know

This module was designed and written with you in mind. It is here to help you master the Logarithmic Functions. The scope of this module permits it to be used in many different learning situations. The language used recognizes the diverse vocabulary level of students. The lessons are arranged to follow the standard sequence of the course. But the order in which you read them can be changed to correspond with the textbook you are now using.

After going through this module, you are expected to:

1. Graph Logarithmic Function
2. Determine the domain and Range of a Logarithmic Function
3. Determine the Intercepts, Zeroes and Asymptotes of a Logarithmic Function

What I Know

Choose the letter of the best answer. Write the chosen letter on a separate sheet of paper.

1. Solve for y; y = log2 ( x+1) if x= A. y=0 B. y=1 C. y=1/
2. What do you think is the inverse of exponential function? A. Logarithm Function B. Rational Function C. Irrational Function
3. If translated to logarithmic function, how does 2x=y transform? A.𝑥 = 𝑙𝑜𝑔2𝑦 B.𝑦 = 𝑙𝑜𝑔2𝑥 C.2 = 𝑙𝑜𝑔𝑥𝑦
4. How do you describe the graph of this function y=log3x? A.The graph of the function is increasing B.The graph of the function is decreasing C.The graph of the function does not exist

5. How do you describe the graph of this function y=log1/2x? A.The graph of the function is increasing B.The graph of the function is decreasing C.The graph of the function does not exist
6. The function 𝑦 = 𝑙𝑜𝑔 2 𝑥 has the range of a. set of negative real numbers c. set of positive real numbers b. set of real numbers d. 0
7. In the function 𝑓(𝑥) = 𝑙𝑜𝑔 (𝑥 + 1 ), the domain is a. {𝑥 ⎸𝑥 > 1} b. {𝑥 ⎸𝑥 > 2} c. {𝑥 ⎸𝑥 > −1} d. {𝑥 ⎸𝑥 > 0}
8. In the function 𝑓(𝑥) = 𝑙𝑜𝑔 (𝑥 + 1 ), the range is a. {𝑥 ⎸𝑥 > 1} b. {𝑥 ⎸𝑥 > 2} c. all real numbers d. {𝑥 ⎸𝑥 > 0}

9. Determine the domain of f(x) = log (2x+2)

a. {𝑥 ⎸𝑥 > 1} b. {𝑥 ⎸𝑥 > 2} c. {𝑥 ⎸𝑥 > −1} d. {𝑥 ⎸𝑥 > 0}

10. In a logarithmic function given by y = - 3log (x-4), determine the domain. a. (-∞,4) b. (4, ∞) c. (-4,∞) d. (-∞,4)
11. In the function f(x) = log5x, the Asymptote is a. x=0 b. x=1 c. 0 d. 1
12. In the function f(x)= log2x, the x-intercept is a. 0 b. 1 c. 2 d. none
13. In the equation y=3log2x-5, which of the following is correct? a. The x-intercept is - 3 4 2 , y-intercept is none, Asymptote is x= 0. b. The x-intercept is 3 4 2 , y-intercept is 5, Asymptote is x= 5. c. The x-intercept is 3 4 2 , y-intercept is 5, Asymptote is x= -5. d. The x-intercept is 3 4 2 , y-intercept is none, Asymptote is x= 0
14. The Asymptote of the logarithmic function is always a. Horizontal Asymptote c. x-intercept b. Vertical Asymptote d. y-intercept
15. The ordinate of the point where the graph of the logarithmic function passes through the y-axis is a. Horizontal Asymptote c. x-intercept b. Vertical Asymptote d. y-intercept

What is It

A. Graphing a Logarithmic Function

Example no.1 Sketch the graph 𝑦 = 𝑙𝑜𝑔 2 𝑋

Solution: STEP 1: Construct a table of values of ordered pairs for the given function. A table of values for 𝑦 = 𝑙𝑜𝑔 2 𝑋 is as follows:

x 1 16

y -4 -3 -2 -1 0 1 2 3

Assuming the value of X

If 𝑥 = 2

𝑦 = 𝑙𝑜𝑔 2 𝑋

𝑦 = 𝑙𝑜𝑔 22 substitute 2 to x

2 𝑦^ = 2 transform logarithmic function to exponential function

𝑦 = 2^1 solve for y

𝑦 = 1 write this value in the table above

Assuming the value of y

If y=

𝑦 = 𝑙𝑜𝑔 2 𝑋 0 = 𝑙𝑜𝑔 2 𝑋 substitute 0 to y

20 = 𝑥 transform logarithmic function to exponential function

1 = 𝑥 write this value in the table above Take note

TAKE NOTE:

You can use any of the given technique above (assuming y or assuming x)

STEP 2: Plot the points found in the Table, and connect them using a smooth curve.

OBSERVATION:

It can be observed that function is defined only for x>0. The function is strictly increasing and attains all real values. As x approaches 0 from the right, the function decreases without bound, i.e., the line x = 0 is a vertical asymptote.

B. DOMAIN AND RANGE

Usually, a logarithm consists of three parts. Let us come to name of those three parts with an example. 𝑙𝑜𝑔 10 𝐴 = 𝐵

In the above logarithmic function, 10 is called the base, A is called as argument and B is called an answer.

Domain of Logarithmic Function

A logarithmic Function is defined only for positive values of arguments. If the logarithmic function is 𝑦 = 𝑙𝑜𝑔 (^) 𝑏𝑥 ,then the domain of the logarithmic function is x > 0 or (0,+∞)

In the logarithmic function 𝑦 = 𝑙𝑜(𝑥), the argument is ‘x’ and it must always be a positive value. So, the values of x must be greater than zero. Therefore, the domain of the logarithmic function is x>0 or (0, ∞).

For example, consider f(x) = log 4 (2x-3). This function is defined for any values of x such that the argument, in this case 2x-3, is greater than zero. To find the domain, we set up an inequality and solve for x:

2x-3>0 Show the argument greater than zero 2x>3 Add 3 to both sides x>1.5 Divide by 2

when y=0. Hence the x-intercept is 2. We will solve for x if y=0.

0 = 𝑙𝑜𝑔 5 (𝑥 − 1) ( we let y=0)

50 = (𝑥 − 1) (Using the law of Log that 𝑏𝑦^ = 𝑥, 𝑦 = 𝑙𝑜𝑔𝑏 𝑥 1 = (𝑥 − 1) (a^0 = 1, if a is not zero) 1 + 1 = 𝑥 (addition property of equality) 2=x. Thus, at 𝑦 = 0, 𝑥 = 2. Therefore the x-intercept is 2. Consequently, the zero of the logarithmic function 𝑓(𝑥) = 𝑙𝑜𝑔 5 (𝑥 − 1) is 2.

y-intercept: viewed and tabled

The graph beside is from 𝑦 = 𝑙𝑜𝑔 2 (𝑥 + 2). The yintercept is 1 because the graph passes the y-axis at point (0,1). The shortest distance (distance vector) from (0,1) to the origin is also 1 unit. The graph passes through the points (-1,0), (0,1), and (2,2). The point (0,1) has an ordinate 1 and it is the value of y when x=0.

We have 𝑦 = 𝑙𝑜𝑔 2 (𝑥 + 2) (Given)

𝑦 = 𝑙𝑜𝑔 2 (0 + 2) (Set x=0 to get y) 𝑦 = 𝑙𝑜𝑔 2 (2) (using the property logaa = 1)

𝑦 = 1 (or 2 𝑦^ = 2^1 , hence y= 1).

What’s More

Activity 1. Graphing logarithmic Function

Fill the table with the correct values of the given function and graph it.

X

Y -3 -2 -2 0 1 2 3

𝟒

X

Y -3 -2 -2 0 1 2 3

𝟒

x

X

Y -3 -2 -2 0 1 2 3

Activity 2 Domain and Range

Given the following logarithmic functions, determine the domain and range.

1. y = log (2x)

2. y = log (x-5)

3. y = log (4x + 4)

Activity 3 x and y- intercept, Zeroes and Asymptote Determine the intercepts (x-intercept and y-intercept), Zeroes, and Asymptote of the following items. Show your solution for items 4 to 7.

4. y= log2(x+1)
5. y= ln(x-2)
6. y = log (x+1)
7. y= -log 3 (x-3)

What I Have Learned

Based on your understanding of our lesson, fill the blank with the correct words.

1. ________________ is the inverse of Exponential Function.
2. It can be observed that the logarithmic function y =logb X (b>0).The function is strictly ____________, and attains all real values.

2. What do you think is the inverse of exponential function? A. Logarithm Function B. Rational Function C. Irrational Function
3. If translated to logarithmic function, how does 2x=y transform? A.x=log2y B.y=log2x C.2=logxy
4. How do you describe the graph of this function y=log3x? A.The graph of the function is increasing B.The graph of the function is decreasing C.The graph of the function does not exist
5. How do you describe the graph of this function y=log1/2x? A.The graph of the function is increasing B.The graph of the function is decreasing C.The graph of the function does not exist
6. The function 𝑦 = 𝑙𝑜𝑔 2 𝑥 has the range of a. set of negative real numbers c. set of positive real numbers b. set of real numbers d. 0
7. In the function f(x) = log (x+1), the domain is a. {x ⎸x>1} b. {x ⎸x>2} c. {x ⎸x>-1} d. {x ⎸x>0}
8. In the function f(x)= log (x+1), the range is a. {x ⎸x>1} b. {x ⎸x>2} c. all real numbers d. {x ⎸x>0}

9. Determine the domain of f(x) = log (2x+2)

a. {x ⎸x>1} b. {x ⎸x>2} c. {x ⎸x>-1} d. {x ⎸x>0}

10. In a logarithmic function given by y = - 3log (x-4), determine the domain. a. (-∞,4) b. (4, ∞) c. (-4,∞) d. (-∞,4)
11. In the function f(x) = log5x, the Asymptote is a. x=0 b. x=1 c. 0 d. 1
12. In the function f(x)= log2x, the x-intercept is a. 0 b. 1 c. 2 d. none
13. In the equation y=3log2x-5, which of the following is correct? a. The x-intercept is - 3 4 2 , y-intercept is none, Asymptote is x= 0. b. The x-intercept is 3 4 2 , y-intercept is 5, Asymptote is x= 5. c. The x-intercept is 3 4 2 , y-intercept is 5, Asymptote is x= -5. d. The x-intercept is 3 4 2 , y-intercept is none, Asymptote is x= 0
14. The Asymptote of the logarithmic function is always a. Horizontal Asymptote c. x-intercept b. Vertical Asymptote d. y-intercept
15. The ordinate of the point where the graph of the logarithmic function passes through the y-axis is a. Horizontal Asymptote c. x-intercept b. Vertical Asymptote d. y-intercept

Additional Activity

1. Graph the function f(x)= log2 (x-1)+2. Then determine the domain and range of the function.
2. If the rate of transmission of an additional case of covid-19 is 0.3 per day starting February 15, 2020 with three cases, that is f(x) = log1.3 (x) + 3, after how many days it will become 109 Million cases?

References

1. Department of Education-Bureau of Learning Resources (DepEd-BLR) (2016) General Mathematics Learner’s Material. Lexicon Press Inc., Philippines
2. Department of Education-Bureau of Learning Resources (DepEd-BLR) (2016) General Mathematics Teacher’s Material. Lexicon Press Inc., Philippines
3. Jose-Dilao, S. & Orines, F.B. (2009). Advanced Algebra, Trigonometry and Statistics. SD Publications, Inc., Quezon City, Philippines
4. Fernandez, P.L, et. Al. (2007). A Course in Freshman Algebra. Ateneo de manila University Press, Quezon City, Philippines

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Department of Education – SOCCSKSARGEN

Learning Resource Management System (LRMS)

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Telefax No.: (083) 2288825/ (083) 2281893

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DISCLAIMER

This Self-learning Module (SLM) was developed by DepEd SOCCSKSARGEN with

the primary objective of preparing for and addressing the new normal. Contents

of this module were based on DepEd’s Most Essential Learning Competencies

(MELC). This is a supplementary material to be used by all learners of Region

XII in all public schools beginning SY 2020 - 2021. The process of LR

development was observed in the production of this module. This is version 1.0.

We highly encourage feedback, comments, and recommendations.