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General Mathematics – Grade 11 Alternative Delivery Mode Quarter 1 Week 8- Module 5: Logarithmic Functions First Edition, 2020
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11
General Mathematics
Quarter 1 – Module 5: Logarithmic Functions
Department of Education - Republic of the Philippines
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Module
5 L O G A R I T H M I C F U N C T I O N S
What I Need to Know
In the previous module, we have learned about exponential functions and its applications
in real life. Logarithmic function is simply the inverse of an exponential function. It is mainly
used, but not limited to, earthquake intensity measurement, acidic measurement of solutions (pH
value), sound intensity measurement and expressing larger values.
This module will help you understand the key concepts of logarithmic functions and apply
these concepts to formulate and solve real-life problems with precision and accuracy.
After finishing the module, you should be able to answer the following questions:
a. How to distinguish logarithmic functions, equations, and inequalities?
b. How to graph logarithmic functions using its intercepts, zeroes and asymptotes?
c. How to solve problems related to logarithmic functions, equations, and inequalities?
In this module, you will examine the aforementioned questions when you study the
following lessons:
Lesson 1: Introduction to Logarithms
Lesson 2: Logarithmic Functions, Equations, and Inequalities
Lesson 3: The Logarithmic Equation and Inequality
Lesson 4: The Logarithmic Function and its Graph
Find out what you already know about this module by taking the pre-test.
What I Know (Pre-Assessment)
Direction: Write the letter that corresponds to the best answer on your answer sheet.
1. Express 271
3⁄ = 3 in logarithmic form.
A. log3 27 = 3 B. log1
3
3 = 27 C. log27 3 =1
3 D. log3 3 = 27
2. Solve for x given the equation, log𝑥 81 = 4.
A. 3 B. 9 C. 20.25 D. 324
3. Evaluate log𝑚 𝑚2𝑛.
A. n B. 𝑛2 C. mn D. 2𝑛
4. Evaluate log2 45.
A. 4 B. 5 C. 7 D. 10
5. Which of the following statements is true?
A. The domain of a transformed logarithmic function is always {𝑥 ∈ 𝑅}
B. Vertical and horizontal translations must be performed before horizontal and vertical
stretches/compressions.
C. A transformed logarithmic function always has a horizontal asymptote.
D. The vertical asymptote changes when a horizontal translation is applied.
6. Which of the following is NOT a strategy that is often used to solve logarithmic equations?
A. Express the equation in exponential form and solve the resulting exponential equation.
B. Simplify the expressions in the equation by using the laws of logarithms.
C. Represent the sums or differences of logs as single logarithms.
D. Square all logarithmic expressions and solve the resulting quadratic equation.
7. Solve for x given the equation 52−𝑥 =1
125.
A. 5
3 B. −1 C. 5 D.
7
3
8. Solve for x given the equation log (3x +1) = 5.
A. 4
3 B. 8 C. 300 D. 33, 333
9. Solve for x given the equation log𝑥 8 = −1
2
A. −64 B. −16 C. 1
64 D. 4
10. Graph 𝑦 = log (x +1) + 7
A.
C.
B.
D.
11. Solve for x given the inequality log3(2𝑥 − 1) > log3(𝑥 + 2)
A. (−3, +∞) B. (3, +∞) C. (−∞, −3) D. (−∞, 3)
12. Solve for x given the inequality −2 < log 𝑥 < 2
A. (−125, 0) B. (0, 125) C. [−125, 0] D. [0, 125]
13. What is the domain of the function, 𝑦 = log0.25(𝑥 + 2)?
A. {𝑥 ∈ 𝑅} B. {𝑥|𝑥 > 0} C. {𝑥|𝑥 > 2} D. {𝑥|𝑥 > −2}
14. What is the range of the function, 𝑦 = log0.25(𝑥 + 2)?
A. {𝑦 ∈ 𝑅} B. {𝑦|𝑦 > 0} C. {𝑦|𝑦 > 2} D. {𝑦|𝑦 > −2}
15. Which of the following does NOT describe the graph of a function in the form 𝑦 = 𝑎 ∗
log𝑏(𝑥 − 𝑐) + 𝑑?
A. The value of a determines the stretch or shrinking of the graph.
B. The value of b determines whether the graph is small or big
C. The value of c determines the horizontal shift of 𝑦 = 𝑎 ∗ log𝑏 𝑥
D. The value of d determines the vertical shift of 𝑦 = 𝑎 ∗ log𝑏 𝑥
How was your performance in the pre-assessment? Were you able to answer all the
problems? Did you find difficulties in answering them? Are there questions familiar to you?
In this module, you will have the following targets:
Demonstrate understanding on the key concepts of logarithmic functions.
Formulate and solve real-life problems involving logarithmic functions with -
precision and accuracy.
Lesson
1 Introduction to Logarithmic Functions
What’s In
Since the previous module is closely related to this module, let’s start this lesson by
reviewing exponents. As you go through this module, keep in mind this question: How is
exponential function related to logarithmic function?
What’s New Activity 1: FIND MY PAIR
Description: This activity will enable you to recall exponents.
Directions: Match the exponential expressions in Column A to its corresponding value of x in
Column B. Write the letter that corresponds to your answer in your answer sheet.
Column A Column B
_____1. 52 = 𝑥
_____2. 33 = 𝑥
_____3. 61 = 𝑥
_____4. 7−2 = 𝑥
_____5. 9−2 = 𝑥
_____6. 8𝑥 =1
64
_____7. 11𝑥 = 121
_____8. 4𝑥 = 1,024
_____9. 2𝑥 = 1
_____10. 10𝑥 = 1,000
A. 27
B. −2 C. 25
D. 5
E. 1
81
F. 2
G. 0 H. 6
I. 3
J. 1
49
What Is It
Direction: Answer the following questions. Write your answers on a separate sheet.
1. What is the main function of the exponents?
2. What have you observed in items 1 to 5 and 6 to 10 in terms of x?
3. How did you answer items 6 to 10? Explain your answer.
In Activity 1, items 6 to 10 have missing exponents. Aside from observation, there a
mathematical way to present (rewrite) the expressions through logarithms, the inverse process of
exponentiation.
Think of a logarithm of x to the base b and power p. That is, if 𝒃𝒑 = 𝒙 then 𝐥𝐨𝐠𝒃 𝒙 = 𝒑
For example,
(a) 52 = 25 is written as log5 25 = 2.
(b) 7−2 =1
49 is written as log7
1
64 = − 2
(c) 33 = 27 is written as log3 27 = 8
Common logarithms are logarithms with base 10, the base is usually omitted when
writing common logarithms. This means that 101 = 10 is written as log 10 = 1 and 102 = 100
is written as log 100 = 2 and so on.
Natural logarithms are logarithms with base e (which is approximately 2.71828 as
mentioned in the previous module). This means that log𝑒 𝑥 can be written as ln 𝑥.
What’s More Activity 2: REWRITE ME!
Description: This activity will enable you to rewrite exponential expressions to logarithmic
expressions and vice versa.
Directions: Write the letter that corresponds to your answer in your answer sheet.
A. Rewrite the following exponential
expressions to logarithmic expressions.
1. 53 = 25
2. 91
2 = 3
3. 2−2 =1
4
4. 105 = 100,000
5. 𝑒4 ≈ 54.598
B. Rewrite the following logarithmic
expressions to exponential expressions.
1. log 1000 = 3
2. log4 16 = 2
3. 4 = log2 16
4. ln 20 ≈ 3
5. log4 64 = 3
Application
Logarithms Real-Life Situations
Your goal in this section is to take a closer look at the real-life applications and problems
involving logarithmic functions.
I. Earthquake Magnitude on a Richter Scale
The magnitude R of an earthquake is given by:
𝑅 =2
3log
𝐸
104.40
where E (in joules) is the energy released by the earthquake (the quantity 104.40 is the energy
released by a very small reference earthquake)
Example:
Suppose that an earthquake released approximately 1012 joules of energy. (a) What is its
magnitude? (b) How much more energy does this earthquake release than by the reference
earthquake?
Solution:
(a)
(b)
𝑹 =2
3log
𝐸
104.40
𝑹 =2
3log
1012
104.40
𝑹 ≈ 𝟓. 𝟏
Magnitude 5 is described as STRONG.
Strong shaking and rocking felt throughout building. Hanging objects swing violently.
1012
104.40= 107.6 ≈ 𝟑𝟗𝟖𝟏𝟎𝟕𝟏𝟕
The earthquake released 39810717 times more energy than the reference earthquake.
What’s New Activity 3: PREPARE TO PREPARE!
Description: This activity will enable you to make a plan whenever an earthquake occurs.
Directions: List down all of the things that you will do before, during and after an earthquake.
II. Sound Intensity
In acoustics, the decibel (dB) level of a sound is
𝑫 = 10 log𝐼
10−12
where I is the sound intensity in watts/𝑚2 (the quantity 10−12 watts/𝑚2 is the least
audible sound a human can hear.
Example:
The decibel level of sound in an office is 10−6 watts/𝑚2. (a) What is the corresponding
sound intensity in decibels? (b) How much more intense is this sound than the least
audible sound a human can hear?
Solution:
a.
EARTHQUAKE RISK REDUCTION AND RECOVERY
Before an Earthquake During an Earthquake After an Earthquake
𝑫 = 10 log𝐼
10−12
𝑫 = 10 log10−6
10−12
𝑫 = 𝟔𝟎 𝒅𝑩
60-85 dB is described as Intrusive.
Examples are vacuum cleaner, washing machine, average city traffic and television.
b.
Enrichment Activity
Activity 4: THINK- PAIR- SHARE
I. Direction: Answer the following questions. If yes, given an example of an answer. If no, explain
why not.
1. Is it possible for the base of a logarithm to equal a negative number?
2. Is it possible for the base of a logarithm to equal zero?
3. Does log𝑥 0 have an answer?
4. Does log𝑥 1 have an answer?
5. Does log𝑥 𝑥5 have an answer?
II. Direction: Answer the following problems. Show a neat and complete solution.
1. An earthquake in Albay released approximately 1018 joules of energy. (a) What is its
magnitude? (b) How much more energy does this earthquake release than by the reference
earthquake?
2. Suppose you have seats to a concert featuring your favorite musical artist. Calculate the
approximate decibel level associated if a typical concerts’ sound intensity is 10−2 W/𝑚2.
10−6
10−12= 106 ≈ 𝟏𝟎𝟎, 𝟎𝟎𝟎
The sound is 100, 000 times more intense than the least audible sound a human can hear.
Lesson
2 Logarithmic Functions, Equations & Inequalities
What’s In
Before this lesson starts, we shall begin the day by discussing the answers to the
enrichment activity guide questions from the previous module. This will serve as a refresher of
what had been already discussed.
What’s New Activity 1: WHICH IS WHICH?
Description: This activity will enable you to distinguish functions, equations and inequalities.
Directions: In your answer sheet, identify if the given is a function, equation or inequality by
writing F, E and I respectively.
1. 𝑦 = 𝑥2 + 1
2. 2𝑥 − 10 > 𝑥 + 3
3. 𝑥2 + 𝑦2 = 1
4. 3𝑥 − 4 = 𝑥 + 10
5. 5𝑥4 + 3 = 5 − 𝑥10
6. log2 𝑥 ≤ 5
7. ln 𝑥 = 𝑦
8. 5 + log3 9 = 7
9. log𝑒 𝑥 + 1 = 2
10. 𝑦 = 2 log4 𝑥
What Is It
Direction: Answer the following questions. Write your answers on a separate sheet.
1. How were you able to identify which given is a function? An
equation? An inequality?
2. What indicators have you noticed in inspecting the given?
3. What are the difficulties you have encountered in doing this activity?
In the previous activity, you have encountered familiar mathematical terms namely
function, equation and inequality. Now let us define these terms with logarithms.
Logarithmic Function Logarithmic Equation Logarithmic Inequality
Definition A function involving
logarithms
An equation involving
logarithms
An inequality
involving logarithms
Example 𝑔(𝑥) = log3 𝑥 log𝑥 2 = 4 ln 𝑥2 > (ln 𝑥)2
What’s More Activity 2: CLASSIFY ME!
Direction: In your answer sheet, write each given in its corresponding column: logarithmic
functions, equations or inequalities.
Logarithmic Function Logarithmic Equation Logarithmic Inequality
log9 1 = log9(1 − 4𝑥) 𝑙𝑜𝑔(3𝑥 − 2) ≤ 2 −ln(1 − 2x) + 1 = g(x) 𝑙𝑜𝑔9 𝑦 + 5 < 20
𝑦 = log𝑥 3 + 5 𝑙𝑜𝑔𝑥 + 𝑙𝑜𝑔(𝑥 − 3) = 1 𝑙𝑜𝑔 32 > 5 𝑓(𝑥) = 𝑙𝑜𝑔 − 𝑥
Enrichment Activity Activity 3: LOGARITHMS CONCEPT MAP
Description: This activity will enable you to master the skill of distinguishing logarithmic
functions, equations and inequalities.
Directions: In your answer sheet, create a concept map of the types of logarithms. This concept
map should show the definition of each type along with 5 examples of each.
Lesson
3 The Logarithmic Equation & Inequality
What’s In Activity 1: 3-2-1 CHART
Before we begin this lesson, let us begin with a simple knowledge check-up.
Description: In this activity, you will be asked to complete the 3-2-1 Chart regarding the previous
lesson on distinguishing logarithmic functions, equations and inequalities.
3-2-1 CHART
Three things I found out:
1.
2.
3.
Two interesting things:
1.
2.
One question I still have:
1.
What’s New Activity 2: THE SECRET MESSAGE
Description: This activity will enable you to solve logarithmic equation.
Directions: Decode the secret message by solving for the value of x in each given below. Write
the corresponding letter of the given to the blank which contain its answer.
T A ! I U
log4 𝑥 = 2 log𝑥 27 = 3 log2
𝑥
3= 4 log3 𝑥 = 4 ln 𝑥 = 3
N S M F H
log2 𝑥 = 5 log𝑥 16 = 4 log3 𝑥 = −2 log 𝑥 = 3 log16 𝑥 =1
2
The secret message:
𝟏
𝟗 3 16 4 81 2 1,000 20.09 32 48
What Is It
Direction: Answer the following questions. Write your answers on a separate sheet.
1. What were the steps you applied in answering each item?
2. What are the difficulties you have encountered? Explain?
I. SOLVING LOGARITHMIC EQUATIONS
Recall that a logarithm of x to the base b and power p written as 𝐥𝐨𝐠𝒃 𝒙 = 𝒑 is an inverse
of the exponential function 𝒃𝒑 = 𝒙 or 𝒙 = 𝒃𝒑. To solve for any logarithmic equation, start with
rewriting logarithmic form to exponential form.
Examples:
(By Rewriting to Exponential Form)
1. log4 𝑥 = 2
If log4 𝑥 = 2 then, 𝑥 = 42
𝒙 = 𝟏𝟔
2. log9 𝑥 =1
2
If log9 𝑥 =1
2 then, 𝑥 = 9
1
2
𝑥 = √9
𝒙 = 𝟑
3. log2𝑦
3= 4
If log2𝑦
3= 4 then,
𝑦
3= 24
𝑦
3= 16
𝑦 = 16 ∗ 3
𝒚 = 𝟒𝟖
4. log (2𝑥 + 1) = 2
If log (2𝑥 + 1) = 2 then, 2𝑥 + 1 = 102
2𝑥 + 1 = 100
2𝑥 = 100 − 1
2𝑥 = 99
𝒙 =𝟗𝟗
𝟐
(By Using One-to-One Property)
One-to-One Property of Logarithmic Function
For any logarithmic function 𝑓(𝑥) = log𝑏 𝑥, if 𝐥𝐨𝐠𝒃 𝒖 = 𝐥𝐨𝐠𝒃 𝒗 then 𝒖 = 𝒗
5. log4(2x) = log4 10
If log4(2x) = log4 10 then, 2x = 10
x =10
2
𝐱 = 𝟓
6. loge(3x + 1) = loge(10)
If loge(3x + 1) = loge(10) then, 3x + 1 = 10
3x = 10 − 1
3x = 9
𝐱 = 𝟑
7. log (4x − 3) = log (2x + 5)
If log (4x − 3) = log (2x + 5) then, 4x − 3 = 2x + 5
4x − 2x = 5 + 3
2x = 8
x =8
2
𝐱 = 𝟒
(By Using the Laws of Logarithms)
Laws of Logarithmic Equations
Law Examples
log𝑏(𝑢𝑣) = log𝑏 𝑢 + log𝑏 𝑣 log7(73 ∙ 78) = log7 73 + log7 78
log𝑏 (𝑢
𝑣) = log𝑏 𝑢 − log𝑏 𝑣 log7 (
45
7) = log7 45 − log7 7
log𝑏(𝑢𝑛) = 𝑛 log𝑏 𝑢 log7(75) = 5 log7 7
8. log x + log(x − 3) = 1
If log x + log(x − 3) = 1 then, log (x)(x − 3) = 1
log (x)(x − 3) = 1
log (x2 − 3𝑥) = 1
x2 − 3𝑥 = 101
x2 − 3𝑥 − 10 = 0
(𝑥 − 5)(𝑥 + 2) = 0
𝒙 = −𝟐, 𝟓
9. log3(x + 25) − log3(x − 1) = 3
If log3(x + 25) − log3(x − 1) = 3 then, log3(x + 25)
(𝑥−1)= 3
(x + 25)
(𝑥−1)= 33
(x + 25)
(𝑥−1)= 27
x + 25 = 27𝑥 − 27
x − 27x = −27 − 25
−26x = −52
𝐱 = 𝟐
10. 2logx = log 2 + log(3x − 4)
If 2logx = log 2 + log(3x − 4)then, log x 2 = log (2) (3x − 4)
x 2 = (2) (3x − 4)
x 2 = 6x − 8
x 2 − 6x + 8 = 0
(𝑥 − 4)(𝑥 − 2) = 0
𝒙 = 𝟐, 𝟒
What’s More
Exercises on Logarithmic Equation
Direction: In your answer sheet, provide the solution and the answer to the following logarithmic
equations.
1. log2 𝑥 = 5
2. log3(𝑥2 + 2) = 3
3. log7(3𝑥) = log7(5𝑥 − 8)
4. log2(4𝑥) − log2(𝑥 − 5) = log2 8
5. log(𝑥2 − 2) + 2 log 6 = log 6𝑥
II. SOLVING LOGARITHMIC INEQUALITY
The second half of this lesson is all about logarithmic inequalities. But before jumping into
solving logarithmic inequalities, let us first have this activity.
What’s New Activity 3: TRACK THE TREND!
Description: This activity will enable you to complete a table of a given and observe its trend.
Directions: Complete the table below mentally.
x 1
8
1
4
1
2 1 2 4 8
log12
𝑥
x 1
8
1
4
1
2 1 2 4 8
log2 𝑥
What Is It
Direction: Answer the following questions. Write your answers on a separate sheet.
1. The base of the first logarithm expression is ½ which is in between 0 and
1. What do you notice with the value log1
2
𝑥 of as x increases?
2. The base of the second logarithm expression is 2 which is greater than 1.
What do you notice with the value log2 𝑥 of as x increases?
We can generalize the observations we made:
The direction of the inequality is based on whether the base b is greater than 1 or lesser than 1.
So, given the logarithmic expression log𝑏 𝑥;
If 𝟎 < 𝒃 < 𝟏, then 𝑥1 < 𝑥2 if and only if log𝑏 𝑥1 > log𝑏 𝑥2
If 𝒃 > 𝟏, then 𝑥1 < 𝑥2 if and only if log𝑏 𝑥1 < log𝑏 𝑥2
Simply means that if the base b is greater than 1, we will retain the inequality symbol of the given.
Otherwise, we will use the opposite symbol.
Examples:
1. log5(3𝑥 − 1) ≤ 1
Since 𝑏 > 1, then log5(3𝑥 − 1) ≤ 1 Retain the symbol
3𝑥 − 1 ≤ 51
3𝑥 ≤ 5 + 1
3𝑥 ≤ 6
𝒙 ≤ 𝟐
Hence, the solution is, [2, +∞)
2. log3(2𝑥 − 1) > log3(𝑥 + 2)
Since 𝑏 > 1, then log3(2𝑥 − 1) > log3(𝑥 + 2) Retain the symbol
2𝑥 − 1 > 𝑥 + 2 One to One Property
2𝑥 − 𝑥 > 2 + 1
𝒙 > 𝟑
Hence, the solution is, (3, +∞)
3. log1
2
(2x + 3) > log1
2
(3x)
Since 0 < 𝑏 < 1, then log1
2
(2x + 3) < log1
2
(3x)
2x + 3 < 3x
2x − 3x < −3
−x < −3
𝐱 < 𝟑
Hence, the solution is, (0, 3) since all logarithms must be positive.
What’s More
Exercises on Logarithmic Inequality
Direction: In your answer sheet, provide the solution and the answer to the following logarithmic
inequalities.
1. log4 𝑥 < 3
2. log0.5(4𝑥 + 1) < log0.5(1 − 4𝑥)
3. log3(1 − 𝑥) ≥ log3(𝑥 + 16 − 𝑥2)
Enrichment Activity
Activity 4: LOGARITHMIC EQUATION AND INEQUALITY MAZE
Description: This activity will enable you to independently solve logarithmic equations.
Directions: Finish the maze by solving the first logarithmic problem and then taking the path
where its answer is written. Do this until you reach the ending point.
Lesson
4 The Logarithmic Function & its Graph
What’s In
Before we proceed with the last lesson of this module, let us check what we have learned
so far through a question and answer activity prepared by the teacher.
What’s New Activity 1: HELP ME GRAPH!
Description: We have learned from the preceded lessons that logarithms can be rewritten in
exponential form. We shall attempt to show that the inverse of exponential
functions is the logarithmic through a graph functions
Directions: With a pair, graph the following functions on the same Cartesian plane.
𝒚 = 𝟐𝒙
x 1
16
1
8
1
4
1
2 1 2 4 8
y
𝒚 = 𝐥𝐨𝐠𝟐 𝒙
x 1
16
1
8
1
4
1
2 1 2 4 8
y
What Is It
Direction: Answer the following questions. Write your answers on a separate sheet.
1. What is the trend of the graph of 𝑦 = 2𝑥?
2. What is the trend of the graph of 𝑦 = log2 𝑥?
3. Compare the two graphs and state your observations.
What’s More
Activity 2: SKETCH TO COMPARE!
Description: This activity will enable you to come up with a generalization about the properties
of a logarithmic function.
Directions: With the same pair, graph the logarithmic function and the questions that follows.
𝒚 = 𝐥𝐨𝐠𝟏𝟐
𝒙
x 1
16
1
8
1
4
1
2 1 2 4 8
y
1. What is the trend of the graph of 𝑦 = log1
2
𝑥?
2. Compare the graph of 𝑦 = log1
2
𝑥 from the previously graphed logarithmic
function 𝑦 = log2 𝑥. State your observations.
3. What can we conclude based on the graph of the two functions in terms of their
base?
Properties of a Logarithmic Function:
i. The domain is the set of all positive numbers; or {𝑥 ∈ 𝑅| 𝑥 > 0}. Recall that these
precisely the permitted values of x in the expression log𝑏 𝑥.
ii. The range is the set of all positive real numbers.
iii. It is a one-to-one function. It satisfies the Horizontal Line Test.
iv. The x-intercept is 1. There is no y-intercept.
v. The vertical asymptote is the line 𝑥 = 0 (or the y-axis). There is no horizontal asymptote
Relationship Between the Graphs of Logarithmic and Exponential Functions
Since logarithmic and exponential functions are inverses of each other, their graphs are
reflections of each other about the line 𝑦 = 𝑥 as shown below.
Graphing Logarithmic Function in the Form 𝒚 = 𝒂 𝐥𝐨𝐠𝒃(𝒙 − 𝒄) + 𝒅
The following examples are given to illustrate graphs of transformations of logarithmic functions.
The 𝒂 in 𝒚 = 𝒂 𝐥𝐨𝐠𝒃(𝒙 − 𝒄) + 𝒅
Example:
Sketch the graphs of 𝒚 = 𝐥𝐨𝐠𝟐 𝒙 and 𝒚 = 𝟐 𝐥𝐨𝐠𝟐 𝒙 on the same Cartesian plane and state your
observation. Also, determine the domain, range, vertical asymptote, x-intercept and zero.
Solution:
Since we already have the graph of the 𝑦 = log2 𝑥 from the first activity, let us focus on the other
function.
𝒚 = 𝟐 𝐥𝐨𝐠𝟐 𝒙
x 1
16
1
8
1
4
1
2 1 2 4 8
y −8 −6 −4 −2 0 2 4 6
The graphs are shown below:
Analysis of the Graphs of Both Functions
a) Domain: {𝑥 ∈ 𝑅| 𝑥 > 0}.
b) Range: {𝑦| 𝑦 ∈ 𝑅}.
c) Vertical Asymptote: 𝑥 = 0
d) x-intercept: 1
e) zero: 1
Graphing Transformation:
The two functions have the properties. However the graph of 𝑦 = 2 log2 𝑥 is stretched
compared to the graph of 𝑦 = log2 𝑥.
The 𝒃 in 𝒚 = 𝒂 𝐥𝐨𝐠𝒃(𝒙 − 𝒄) + 𝒅
Example:
Sketch the graphs of 𝒚 = 𝐥𝐨𝐠𝟐 𝒙 and 𝒚 = 𝐥𝐨𝐠𝟏
𝟐
𝒙 on the same Cartesian plane and state your
observation. Also, determine the domain, range, vertical asymptote, x-intercept and zero.
Solution:
We already have a table of values and graphs of both functions. (See Activity 2)
Analysis of the Graphs of Both Functions:
a) Domain: {𝑥 ∈ 𝑅| 𝑥 > 0}.
b) Range: {𝑦| 𝑦 ∈ 𝑅}.
c) Vertical Asymptote: 𝑥 = 0
d) x-intercept: 1
e) zero: 1
Graphing Transformation:
The two functions have the properties. However the graph of 𝑦 = log2 𝑥 is increasing
while the graph of 𝑦 = log1
2
𝑥 is decreasing.
The 𝒄 in 𝒚 = 𝒂 𝐥𝐨𝐠𝒃(𝒙 − 𝒄) + 𝒅
Example:
Sketch the graphs of 𝒚 = 𝐥𝐨𝐠𝟏
𝟒
𝒙 and 𝒚 = 𝐥𝐨𝐠𝟏
𝟒
(𝒙 + 𝟐) on the same Cartesian plane and state your
observation. Also, determine the domain, range, vertical asymptote, x-intercept and zero.
Solution:
𝒚 = 𝐥𝐨𝐠𝟏
𝟒
𝒙
x 1
4 1 4
y 1 0 -1
𝒚 = 𝐥𝐨𝐠𝟏𝟒
(𝒙 + 𝟐)
x −13
4 -1 2
y 1 0 -1
The graphs are shown below:
Analysis of the Graphs:
𝑦 = 𝑙𝑜𝑔1
4
𝑥:
a) Domain: {𝑥 ∈ 𝑅| 𝑥 > 0}.
b) Range: {𝑦| 𝑦 ∈ 𝑅}.
c) Vertical Asymptote: 𝑥 = 0
d) x-intercept: 1
e) zero: 1
𝑦 = 𝑙𝑜𝑔1
4
(𝑥 + 2):
a) Domain: {𝑥 ∈ 𝑅| 𝑥 > −2}.
b) Range: {𝑦| 𝑦 ∈ 𝑅}.
c) Vertical Asymptote: 𝑥 = −2
d) x-intercept: -1
e) zero: -1
Graphing Transformation:
The two graphs are exactly the same in shape and direction. However the graph of 𝑦 =
log1
4
(𝑥 + 2) is shift to the left by 2 units.
The 𝒅 in 𝒚 = 𝒂 𝐥𝐨𝐠𝒃(𝒙 − 𝒄) + 𝒅
Example:
Sketch the graphs of 𝒚 = 𝐥𝐨𝐠𝟑 𝒙 and 𝒚 = 𝐥𝐨𝐠𝟑 𝒙 − 𝟏 on the same Cartesian plane and state your
observation. Also, determine the domain, range, vertical asymptote, x-intercept and zero.
Solution:
𝒚 = 𝐥𝐨𝐠𝟑 𝒙
x 1 3 9
y 0 1 2
𝒚 = 𝐥𝐨𝐠𝟑 𝒙 − 𝟏
x 1 3 9
y -1 0 1
The graphs are shown below:
Analysis of the Graphs:
𝑦 = 𝑙𝑜𝑔3 𝑥:
f) Domain: {𝑥 ∈ 𝑅| 𝑥 > 0}.
g) Range: {𝑦| 𝑦 ∈ 𝑅}.
h) Vertical Asymptote: 𝑥 = 0
i) x-intercept: 1
j) zero: 1
𝑦 = 𝑙𝑜𝑔3 𝑥 − 1:
f) Domain: {𝑥 ∈ 𝑅| 𝑥 > 0}.
g) Range: {𝑦| 𝑦 ∈ 𝑅}.
h) Vertical Asymptote: 𝑥 = 0
i) x-intercept: 3
j) zero: 3
Graphing Transformation:
The two graphs are exactly the same in shape and direction. However the graph of 𝑦 =
log3 𝑥 − 1 is shift downwards by 1 unit.
Graph of 𝒚 = 𝒂 𝐥𝐨𝐠𝒃(𝒙 − 𝒄) + 𝒅
The value of a determines the stretch or shrinking of the graph. Further, is a is negative,
there is a reflection of the graph about the x-axis.
The value of b determines whether the graph is increasing or decreasing.
The value of c determines whether the graph shifts to the left or to the right.
The value of d determines whether the graph shifts upward or downward.
Application
Exercise
Direction: For each of the following functions; (a) use transformations to describe how the graph
is related to an logarithmic function 𝑦 = log𝑏 𝑥 ; (b) sketch the graph, and (c) identify the domain,
range, vertical asymptote, y-intercept, zero.
1. 𝑦 = log𝑥(𝑥 + 3)
2. 𝑦 = log1
3
(𝑥 − 1)
3. 𝑦 = log5 𝑥 + 6
4. 𝑦 = log0.1 𝑥 − 2
5. 𝑦 = log2
5
( 𝑥 − 4) + 2
6. 𝑦 = log6( 𝑥 + 1) + 5
Enrichment Activity
Activity 3: BACK IN TIME!
Direction: Work with a pair and do what is asked in the given below.
Before calculators were invented, people used a table of logarithms to compute for certain
numbers.
Table of Logarithms log 1 = 0 log 2 ≈ 0.3010 log 3 ≈ 0.4771 log 4 ≈ 0.6021 log 5 ≈ 0.6990
log 6 ≈ 0.7782 log 7 ≈ 0.8451 log 8 ≈ 0.9031 log 9 ≈ 0.9542 log 10 ≈ 1
Brainstorm as a pair and decide how exponents and logarithms can be used to approximate the
value of 21/3
51/4 .
What I Know (Post-Assessment) Direction: Write the letter that corresponds to the best answer on your answer sheet.
1. Express 271
3⁄ = 3 in logarithmic form.
A. log3 27 = 3 B. log1
3
3 = 27 C. log27 3 =1
3 D. log3 3 = 27
2. Solve for x given the equation, log𝑥 81 = 4.
A. 3 B. 9 C. 20.25 D. 324
3. Evaluate log𝑚 𝑚2𝑛.
A. n B. 𝑛2 C. mn D. 2𝑛
4. Evaluate log2 45.
A. 4 B. 5 C. 7 D. 10
5. Which of the following statements is true?
A. The domain of a transformed logarithmic function is always {𝑥 ∈ 𝑅}
B. Vertical and horizontal translations must be performed before horizontal and vertical
stretches/compressions.
C. A transformed logarithmic function always has a horizontal asymptote.
D. The vertical asymptote changes when a horizontal translation is applied.
6. Which of the following is NOT a strategy that is often used to solve logarithmic equations?
7. Solve for x given the equation 52−𝑥 =1
125.
A. 5
3 B. −1 C. 5 D.
7
3
8. Solve for x given the equation log (3x +1) = 5.
A. 4
3 B. 8 C. 300 D. 33, 333
9. Solve for x given the equation log𝑥 8 = −1
2
A. −64 B. −16 C. 1
64 D. 4
A. Express the equation in exponential form and solve the resulting exponential equation.
B. Simplify the expressions in the equation by using the laws of logarithms.
C. Represent the sums or differences of logs as single logarithms.
D. Square all logarithmic expressions and solve the resulting quadratic equation.
10. Graph 𝑦 = log (x +1) + 7
A.
C.
B.
D.
11. Solve for x given the inequality log3(2𝑥 − 1) > log3(𝑥 + 2)
A. (−3, +∞) B. (3, +∞) C. (−∞, −3) D. (−∞, 3)
12. Solve for x given the inequality −2 < log 𝑥 < 2
A. (−125, 0) B. (0, 125) C. [−125, 0] D. [0, 125]
13. What is the domain of the function, 𝑦 = log0.25(𝑥 + 2)?
A. {𝑥 ∈ 𝑅} B. {𝑥|𝑥 > 0} C. {𝑥|𝑥 > 2} D. {𝑥|𝑥 > −2}
14. What is the range of the function, 𝑦 = log0.25(𝑥 + 2)?
A. {𝑦 ∈ 𝑅} B. {𝑦|𝑦 > 0} C. {𝑦|𝑦 > 2} D. {𝑦|𝑦 > −2}
15. Which of the following does NOT describe the graph of a function in the form 𝑦 = 𝑎 ∗
log𝑏(𝑥 − 𝑐) + 𝑑?
A. The value of a determines the stretch or shrinking of the graph.
B. The value of b determines whether the graph is small or big
C. The value of c determines the horizontal shift of 𝑦 = 𝑎 ∗ log𝑏 𝑥
D. The value of d determines the vertical shift of 𝑦 = 𝑎 ∗ log𝑏 𝑥
KEY ANSWERS
Pre/Post-Assessment:
1. C 2. A 3. D 4. D 5. D 6. D 7. C 8. D
9. C 10. C 11. B 12. B 13. D 14. A 15. B
Lesson 1
Activity 1
1. C 2. A 3. H 4. J 5. E 6. B 7. F 8. D 9. G 10. I
Activity 2
Part A Part B
1. log5 25 = 3 1. 103 = 1000
2. log9 3 =1
2 2. 42 = 16
3. log21
4= −2 3. 24 = 16
4. log 100,000 = 5 4. 𝑒3 ≈ 20
5. ln 54.598 ≈ 4 5. 43 = 64
Activity 3
Answers may vary.
Activity 4
I.
1. No. This would give us a discontinuous function.
Consider a hypothetical negative base of −4, hence log−4 𝑥 = 𝑝 with the inverse, (−4)𝑝 = 𝑥.
If you raise −4 to a fraction, this does not allow you to have an even denominator since this
would result to an imaginary number. Say, (−4)1
2 = √−4 which is an imaginary number.
2. No. There is no exponent you can put on 0 that won’t give you back a value of 0.
3. No. It is undefined.
4. Yes. The answer is always 0.
5. Yes. The answer is 5.
II.
1. Magnitude 8.8
2. 100 decibels
Lesson 2
Activity 1
1. F 2. I 3. F 4. E 5. E 6. I 7. F 8. E 9. E 10. F
Activity 2
Logarithmic Function Logarithmic Equation Logarithmic Inequality
𝑦 = log𝑥 3 + 5
−ln(1 − 2x) + 1 = g(x) 𝑓(𝑥) = 𝑙𝑜𝑔 − 𝑥
log9 1 = log9(1 − 4𝑥)
𝑙𝑜𝑔𝑥 + 𝑙𝑜𝑔(𝑥 − 3) = 1
log(3𝑥 − 2) ≤ 2
𝑙𝑜𝑔 32 > 5
𝑙𝑜𝑔9 𝑦 + 5 < 20
Activity 3
Answers may vary.
Lesson 3
Activity 1
Answers may vary
Activity 2
The secret message: MATH IS FUN!
Activity 3:
x 1
8
1
4
1
2 1 2 4 8
log12
𝑥 3 2 1 0 -1 -2 -3
x 1
8
1
4
1
2 1 2 4 8
log2 𝑥 -3 -2 -1 0 1 2 3
Activity 4:
Lesson 4
Activity 1
Activity 2
Activity 3:
Let 𝑛 =21/3
51/4 . Then, using the laws of logarithms, log 𝑛 = log21/3
51/4 which can be further written
as log 𝑛 =1
3log2 −
1
4log 5.
Use the table above to approximate log 𝑛:
log 𝑛 =1
3(0.3010) −
1
4log(0.6990) ≈ −0.0744
However, −0.0744 is just the log of 𝑛 −it is not yet the value of 𝑛. That is,
log 𝑛 ≈ −0.0744
Therefore, 𝑛 ≈ 10−0.0744. The value of this number can be found using a logarithm table.
References
1. Senior High School General Mathematics Teaching Guide
2. https://lor.usq.edu.au/usq/file/5a256444-7623-4919-9b14-
c8f288ef5b60/1/B5_Exponential_and_Logarithmic_Functions_Jan14.pdf
3. https://1.cdn.edl.io/fkJUjSVfklY6552aaosSkorqhr8ykoKoyf6MhiL88LTWOmiX.pdf
4. http://www.mathematicsvisionproject.org/uploads/1/1/6/3/11636986/sec3mod2tnh718.pd
f
5. https://math.colorado.edu/math1300/resources/Exercises_LogarithmicFunction.pdf
6. http://www.unm.edu/~efryer/Review%20Files/Worksheets/Exponential%20and%20Log
orithmic%20Functions.pdf
7. https://www.ms.uky.edu/ma109/fall_2016/activities/10_exponential_worksheet_solution.
8. https://maths.mq.edu.au/numeracy/web_mums/module2/Worksheet27/module2.pdf
9. http://misternolfi.com/Courses/Mhf4u0/Logarithms%20Practice%20Test.pdf
10. http://moodle.tbaisd.org/pluginfile.php/68285/mod_resource/content/0/Assessments/Exp
onentialsLogsMCPreTest.pdf
11. https://brilliant.org/wiki/logarithmic-inequalities/#logarithmic-inequalities-same-base
12. https://www.math-exercises.com/equations-and-inequalities/logarithmic-equations-and-
inequalities