General Mathematics Quarter 1 Module 25: Representations of Logarithmic Functions

General Mathematics Quarter 1 – Module 25: Representations of Logarithmic Functions

Senior High School

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CO_Q1_General Mathematic SHS Module 25

General Mathematics Alternative Delivery Mode Quarter 1 – Module 25: Representations of Logarithmic Functions First Edition, 2021

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General Mathematics Quarter 1 – Module 25: Representations of Logarithmic Functions

Senior High School

Introductory Message

This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.

In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.

Thank you.

What I Need to Know

This module focuses on logarithmic functions represented through the table of values, graphs and its transformation, and equation. Logarithmic function as the inverse of the exponential function have properties that are somewhat related to exponential functions. Using the representations of logarithmic functions will give the ideas of how these two functions are related to each other.

After going through this module, you are expected to:

1. solve exponential and logarithmic equation;
2. represent logarithmic function through its table of values, graph, and equation; and
3. sketch the transformation of the graph of logarithmic function.

What I Know

Before studying this module, take this test to determine what you already know about the topic covered. Choose the letter of the best answer. Write the chosen letter on a separate sheet of paper.

1. The graph of the logarithmic function f(x)=logbx is _________ if b > 1.
   a. decreasing
   b. increasing
   c. shifted down
   d. shifted up

   
   

2. The graph of the logarithmic function f(x)=logb(x)+d is shifted ______ if d < 0.
   a. downward
   b. left
   c. right
   d. upward

   
   

3. What is the inverse of the function x = b^y?
   a. b=logxy
   b. x=logby
   c. y=logbx
   d. y=logxb

   
   

4. The graph of the logarithmic function f(x)=logb(x+c) is shifted _______ if c > 0.
   a. downward
   b. left
   c. right
   d. upward

   
   

5. The graph of the logarithmic function f(x)=a logb(x) is stretched if _______.
   a. a>1
   b. a<1
   c. 0 < a < 1
   d. 1 < a < 0

   
   

6. Which of the following is the graph of y = log2x?
   a. [chart: graph of y = log2x, option a; Values are not labeled on the chart.]
   c. [chart: graph of y = log2x, option c; Values are not labeled on the chart.]
   b. [chart: graph of y = log2x, option b; Values are not labeled on the chart.]
   d. [chart: graph of y = log2x, option d; Values are not labeled on the chart.]

   
   

7. The graph of the logarithmic function f(x)=logbx is _________ if 0 < b < 1.
   a. decreasing
   b. increasing
   c. shifted down
   d. shifted up

   
   

8. What is the inverse of the logarithmic function?
   a. exponential
   b. linear
   c. c. polynomial
   d. d. quadratic

   
   

9. Which of the following is the inverse of f(x) = log2x?
   a. y=x2
   b. y=2x
   c. 2y=x
   d. d. x=y2

   
   

10. The graph of the logarithmic function f(x)=logb(x)+d is shifted ______ if d > 0.
    a. down
    b. left
    c. right
    d. up

    
    

11. Which of the following is the graph of y = log1/2x?
    a. [chart: graph of y = log1/2x, option a; Values are not labeled on the chart.]
    c. [chart: graph of y = log1/2x, option c; Values are not labeled on the chart.]
    b. [chart: graph of y = log1/2x, option b; Values are not labeled on the chart.]
    d. [chart: graph of y = log1/2x, option d; Values are not labeled on the chart.]

    
    

12. The graph of the logarithmic function f(x)=a logb(x) is compressed if _______.
    a. a>1
    b. a<1
    c. 0 < a < 1
    d. 1 < a < 0

    
    

13. The graph of the logarithmic function f(x)=logb(x+c) is shifted _______ if c< 0.
    a. down
    b. left
    c. right
    d. up

    
    

14. Which of the following is the graph of y=2logx?
    a. [chart: graph of y = 2logx, option a; Values are not labeled on the chart.]
    c. [chart: graph of y = 2logx, option c; Values are not labeled on the chart.]
    b. [chart: graph of y = 2logx, option b; Values are not labeled on the chart.]
    d. [chart: graph of y = 2logx, option d; Values are not labeled on the chart.]

    
    

15. Which of the following is the graph of y=log (x-2)?
    a. [chart: graph of y = log(x-2), option a; Values are not labeled on the chart.]
    c. [chart: graph of y = log(x-2), option c; Values are not labeled on the chart.]
    b. [chart: graph of y = log(x-2), option b; Values are not labeled on the chart.]
    d. [chart: graph of y = log(x-2), option d; Values are not labeled on the chart.]

    
    

Lesson 1 Representations of Logarithmic Functions

What’s In

In representing logarithmic function through the table of values and graph, it is important to recall your knowledge on how to transform the logarithmic equation to an exponential equation. In doing so, let us start with the following examples on how to change logarithmic equation into exponential equation. This will help you as you go along with this module.

a. log4 64 = 3 b. log36 6 = 1/2 c. log2 (1/4) = −2

In changing logarithmic equation to exponential function, it can be recalled that the logarithmic function y=logb x is the inverse of the exponential function y=b^x and you just need to remember that you are answering the question “To what power must b be raised to obtain the number x?”.

[chart: diagram; logb x = y means b^y = x. Labels: “base”, “power(exponent)” (above logb x = y), “logarithmic form”; “base”, “power(exponent)” (above b^y = x), “exponential form”]

Let us now try the examples given above.

a. log4 64 = 3 4^3 = 64 In this example, b=4, y=3 and x=64

b. log36 6 = (1/2) 36^(1/2) = 6 In this example, b=36, y= and x=6

c. log2 (1/4) = -2 2^-2 = 1/4 In this example, b=2, y= -2 and x=

[transcriber’s note: in the source, the value of y for example b and the value of x for example c are blank in the printed annotations]

Now that you know how to change the logarithmic equation into an exponential function, it is now time to have a glimpse of the graph of a logarithmic function by doing the activity prepared for you.

What’s New

Visualizing the Graphs

Match the function with its graph by writing the letter of your answer on a sheet of paper. (Note: You have to use the available graphing tools online like Desmos, Geogebra, etc.)

[chart: graphs; a–f. Six curve graphs on coordinate grids labeled a to f. Graphs d and f show labeled axes with the numbers 2, 4, 6, 8, 10, 12 on the x-axis.]

____ 1. f(x) = log (x+2) ____ 2. f(x) = log3 x ____ 3. f(x) = log(x-3) + 2 ____ 4. f(x) = log¼ x ____ 5. f(x) = 2log ____ 6. f(x) = log (x-2)

[transcriber’s note: item 5 is printed without an argument]

What is It

In the previous activity, you are going to match the given graphs to their respective equations or functions and try to identify the graphs of a logarithmic function.

After doing the activity, it is time for you to reflect on the following:

1. Do you find difficulty in determining the graph of a function?
2. What can you say axbout the graph of a logarithmic function?
3. How about the properties of logarithmic function f(x)=logb x?

Graph of a Logarithmic Function

Let us begin with the parent function f(x) = logb x. Because every logarithmic function of this form is the inverse of the exponential function with the form f(x) = b^x, their graphs are reflections of each other across the line y = x, as shown below.

[chart: two coordinate-grid panels. Left panel “b > 1”: f(x) = 2^x (exponential curve) and f^-1(x) = log2 x (logarithmic curve), reflected across the line y = x. Right panel “0 < b < 1”: f(x) = (1/2)^x and f^-1(x) = log1/2 x. Values are not labeled on the chart.]

For any real number x and constant b > 0, b ≠ 1, we can see the following characteristics in the graph f(x)=logb x: one-to-one function; vertical asymptote x = 0; domain: (0, ∞) and range: (-∞, ∞); x-intercept: (1,0) and key point (b, 1), y-intercept: none; increasing if b > 1 and decreasing if 0 < b < 1.

Example 1. Sketch the graph of y = log2 x.

Solution:

Step 1: Construct a table of values of ordered pairs for the given function by assigning values for y. Then solve for x. A table of values for y=log2 x is as follows:

x	1/16	1/8	1/4	1/2	1	2	4	8
y	-4	-3	-2	-1	0	1	2	3

Now, let us solve x given y = log2 x where y=-4 from the table.

y = log2 x → Convert into exponential equation log2 x = -4 → b^y = x → You need to rewrite log2 x=y in the form b^y=x. Here, the base is 2 and the exponent is -4. b^y = x → 2^-4 = x → Substitute for b, y, and x in the exponential equation, b^y=x. x = 1/2^4 = 1/16 → Solve for x.

Now, continue with the remaining values of y to complete the table.

Step 2. Plot the points found in the table and connect them using a smooth curve.

Example 2. Sketch the graph of y = log1/2 x.

Solution:

Step 1: Construct a table of values of ordered pairs for the given function by assigning values for y. Then solve for x. A table of values for y = log1/2 x is as follows:

x	16	8	4	2	1	1/2	1/4	1/8	1/16
y	-4	-3	-2	-1	0	1	2	3	4

[transcriber’s note: the base of the logarithm in this example is printed as a subscript fraction, read here as 1/2]

Now, let us solve x given y = log1/2 x where y=-4 from the table.

log1/2 x = y → Convert into exponential equation. log1/2 x = -4 → b^y = x → You need to rewrite x = y in the form b^y=x. Here, the base is ½ and the exponent is -4. b^y = x → (1/2)^-4 = x → Substitute for b, y, and x in the exponential equation, b^y=x. x = 1/(1/2)^4 = 1/(1/16) = 16 → Solve for x.

Now, continue with the remaining values of y to complete the table.

Step 2. Plot the points found in the table and connect them using a smooth curve.

[chart: two coordinate-grid panels. Left: “Plotting of points for y = log1/2 x” (points only). Right: “Graph of y=log1/2 x” (smooth decreasing curve). Values are not labeled on the chart.]

Example 3. Sketch the graph of y = log4 x.

Solution:

Step 1: Construct a table of values of ordered pairs for the given function by assigning values for y. Then solve for x. A table of values for y=log4 x is as follows:

x	1/16	1/4	1	4	16
y	-2	-1	0	1	2

Now, let us solve for x given y = log4 x where y=-2 from the table.

y = log4 x → Convert into exponential equation. log4 x = -2 → b^y = x → Rewrite log4 x=y in the form b^y=x. Here, the base is 4 and the exponent is -2. b^y = x → 4^-2 = x → Substitute for b, y, and x in the exponential equation, b^y=x. x = 1/4^2 = 1/16 → Solve for x.

Now, continue with the remaining values of y to complete the table.

Step 2. Plot the points found in the table and connect them using a smooth curve.

[chart: two coordinate-grid panels. Left: “Plotting of points for y=log3x” (points only). Right: “Graph of y=log4x” (smooth increasing curve). Values are not labeled on the chart.]

[transcriber’s note: the left panel caption is printed as “y=log3x” in the source]

Example 4. Sketch the graph of y = log1/4 x.

Solution:

Step 1: Construct a table of values of ordered pairs for the given function by assigning values for y then solve for x. A table of values for y = x is as follows:

x	16	4	1	1/4	1/16
y	-2	-1	0	1	2

Now, let us solve for x given y = x where y=-2 from the table.

log1/4 x = y → Convert into exponential equation log1/4 x = -2 → b^y = x → You need to rewrite x = y in the form b^y=x. Here, the base is 1/4 and the exponent is -2. b^y = x → (1/4)^-2 = x → Substitute for b, y, and x in the exponential equation, b^y=x. x = 1/(1/4)^2 = 1/(1/16) = 16 → Solve for x.

Now continue with the remaining values of y to complete the table.

Step 2. Plot the points found in the table and connect them using a smooth curve.

[chart: two coordinate-grid panels. Left: “Plotting of points for” (points only; caption ends without a function). Right: “Graph of” (smooth decreasing curve; caption ends without a function). Values are not labeled on the chart.]

[transcriber’s note: both captions are printed incomplete in the source]

In addition to the graphs of the logarithmic function, let us also take a look at how the graph of the parent function y=logb x transform either by shifting to the left or right, shifting up or down, stretches, and compresses.

Graphing Transformations of Logarithmic Functions

Transformation of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, and compress the parent function y = logb(x) without loss of shape.

A. Horizontal Shifts of the Parent Function y = logb (x)

For any constant c, the function f(x) = logb (x+c) ● shifts the parent function y = logb (x) left c units if c > 0. ● shifts the parent function y = logb (x) right c units if c < 0. ● has the vertical asymptote x = – c. ● has domain (-c, ∞). ● has range (-∞, ∞).

[chart: two coordinate-grid panels. Left “Shift left g(x) = logb(x + c)”: curve g(x) = logb(x + c) with vertical asymptote x = -c, labeled points (b – c, 1), (b, 1), (1 – c, 0), (1, 0), and parent curve f(x) = logb(x) with vertical asymptote x = 0; note “The asymptote changes to x = -c.” Right “Shift right h(x) = logb(x – c)”: curve h(x) = logb(x – c) with vertical asymptote x = c, labeled points (b + c, 1), (1 + c, 0), (b, 1), (1, 0), and parent curve f(x) = logb(x) with vertical asymptote x = 0; note “The asymptote changes to x = c.”]

Example 1. Sketch the graph of f(x) = log3(x-2) alongside its parent function.

Solution:

Since the function f(x) = log3(x-2), we notice x + (-2) = x – 2. Thus, c = -2, so c < 0. This means we will shift the function f(x) = log3(x) right 2 units. The vertical asymptote is x = -(-2) or x = 2.

● Consider three key points from the parent function such as (1/3, -1), (1,0), and (3,1) then add 2 to the x coordinates for the new coordinates (7/3, -1), (3,0), and (5,1).

Graph:

[chart: coordinate grid with vertical asymptotes x = 0 (parent) and x = 2 (shifted). Curves: y = log3(x) (parent) labeled with points (1/3, -1), (1, 0), (3, 1); f(x) = log3(x – 2) labeled with points (3, 0) and (5, 1). Values are labeled on the chart as printed.]

B. Vertical Shifts of the Parent Function y = logb(x)

For any constant d, the function f(x) = logb(x) + d. ● shifts the parent function y = logb(x) up d units if d > 0. ● shifts the parent function y = logb(x) down d units if d < 0. ● has the vertical asymptote x = 0.

Example 2. Sketch the graph of f(x) = log3(x) – 2 alongside its parent function.

Solution:

Since the function is f(x) = log3(x) -2, we will notice d = -2. Thus, d < 0. This means we will shift the function f(x) = log3(x) down 2 units. The vertical asymptote is x = 0.

● Consider three key points from the parent function such as (1/3,-1), (1,0), and (3,1) then subtract 2 from the y coordinates for the new coordinates (1/3, -3), (1,-2), and (3,-1).

Graph:

[chart: two-part figure. Top: “Shift up g(x) = logb(x) + d” with curve g(x) = logb(x) + d, asymptote x = 0, labeled points (b^(1-d), 1), (b^-d, 0), (b, 1), (1, 0), and parent f(x) = logb(x). Note: “The asymptote remains x = 0.” Right: “Shift down h(x) = logb(x) – d” with curve h(x) = logb(x) – d, asymptote x = 0, labeled points (b, 1), (1, 0), (b^d, 0), (b^(1+d), 1), and parent f(x) = logb(x). Note: “The asymptote remains x = 0.”]

[chart: coordinate grid with asymptote x = 0. Curves: y = log3(x) (parent) labeled points (1/3, -1), (1, 0), (3, 1); f(x) = log3(x) – 2 labeled points (1/3, -3), (1, -2), (3, -1). Values are labeled on the chart as printed.]

C. Vertical Stretches and Compressions of the Parent Function y=log_b(x)

For any constant b > 1, the function f(x) = a log_b(x)

• stretches the parent function y = log_b(x) vertically by a factor of a if a > 1.
• compresses the parent function y = log_b(x) vertically by a factor of a if 0 < a < 1.
• has the vertical asymptote x = 0.
• has the x-intercept (1,0).

[chart: two-panel diagram; left panel “Vertical Stretch g(x) = a log_b(x), a > 1”, right panel “Vertical Compression h(x) = 1/a log_b(x), a > 1 [?]“. Left panel curves: g(x) = a log_b(x), f(x) = log_b(x). Left panel points: (b^(1/a), 1), (b, 1), (1, 0). Left panel lines: x=0. Left panel notes: “The asymptote remains x = 0.” and “The x-intercept remains (1, 0).” Right panel curves: f(x) = log_b(x), h(x) = ½ log_b(x). Right panel points: (b, 1), (b^2, 1), (1, 0). Right panel line: x=0. Right panel notes: “The asymptote remains x = 0.” and “The x-intercept remains (1, 0).”]

Example 3. Sketch the graph of f(x) = 2 log4(x) alongside its parent function.

Solution:

Since the function f(x) = 2 log4(x), we will notice a = 2. This means we will stretch the function f(x) = log4(x) by a factor of 2. The vertical asymptote is x = 0.

• Consider three key points from the parent function such as (1/4, -1), (1,0), and (4,1) then multiply the y-coordinate by 2 for the new coordinates (1/4, -2), (1,0), and (4,2).

Graph:

[chart: graph of two logarithmic curves on a coordinate plane; labels: x=0 (vertical asymptote), f(x) = 2 log4(x), y = log4(x); points labeled (2, 1), (4, 1), (1, 0)]

What’s More

Activity 1.1

Match It!

Match the following logarithmic functions to its corresponding graph by drawing a line to connect them.

1.) f(x) = log (x-1)+2 2.) f(x) = log2x 3.) f(x) = log x² 4.) f(x) = log (x+3) 5.) f(x) = 3 – ½logx

a.) [illustration: graph with two branches opening upward, vertex near the origin, curves labeled with red arrows] b.) [illustration: graph of a logarithmic curve increasing from a vertical asymptote on the left, with a labeled axis] c.) [illustration: graph of a decreasing curve approaching the vertical axis] d.) [illustration: graph of an increasing logarithmic curve on a grid] e.) [illustration: graph of an increasing logarithmic curve on a grid]

Activity 1.2

Let’s “Pair”

Complete the table of values and find the inverse of the given function. Sketch the graphs of the function and its inverse on a separate sheet of paper.

X	1/1000			1			1000
y = log x		-2	-1		1	2

X	-1	-2	-4	-8	-16	-32
y= log4(-x)

Activity 1.3

Graph the following using table of values:

1. f(x) = log2 (x+3)
2. f(x) = log3 (x) – 1
3. f(x) = ½ log x`
4. f(x) = 4log x
5. f(x) = log3 (x – 2)

What I Have Learned

Fill in the blanks with the correct answer.

• The graph of a logarithmic function has a vertical asymptote at ______.
• The graph of the logarithmic function f(x)=logb(x) is increasing if ______ and decreasing if _______.
• The graph of the function f(x)=logb (x+c) shifted the parent function f(x)=logb(x) to the _____ if c > 0 and to the _____ if c < 0.
• The graph of the function f(x)=logb(x) + d shifted the parent function f(x)=logb(x) upward if _____ and downward if ____.
• The graph of the function f(x)=a logbx _______ the parent function f(x)=logbx if ______ and compresses if ______.

What I Can Do

Answer the problem given below.

Loudness of Sound

The loudness L, in bels (after Alexander Graham Bell), of a sound of intensity I is defined to be

L = log I/I_o

where I_o is the minimum intensity detectable by the human ear (such as the tick of a watch at 20 ft under quiet conditions). If a sound is 10 times as intense as another, its loudness is 1 bel greater than that of the other. If a sound is 100 times as intense as another, its loudness is 2 bels greater, and so on. The bel is a large unit, so a subunit, the decibel, is generally used. For L, in decibels, the formula is

L = 10 log I/I_o

Find the loudness, in decibels, of each sound with the given intensity.

SOUND	INTENSITY
a. Jet engine at 100 ft.	10^14 • I_o
b. Loud rock concert	10^11.5 • I_o
c. Bird calls	10^4 • I_o
d. Normal conversation	10^6.5 • I_o
e. Thunder	10^12 • I_o
f. Loudest sound possible	10^19.4 • I_o

a.) How will you help in reducing noise pollution in your community?

Rubric for rating this activity.

Score	Descriptors
20	The situation is correctly modeled with an exponential and logarithmic function, appropriate mathematical concepts are fully used in the solution and the correct final answer is obtained.
15	The situation is correctly modeled with an exponential and logarithmic function, appropriate mathematical concepts are partially used in the solution and the correct final answer is obtained.
10	The situation is not modeled with an exponential and logarithmic function, other alternative mathematical concepts are used in the solution and the correct final answer is obtained.
5	The situation does not model an exponential and logarithmic function, a solution is presented but has an incorrect final answer.

Assessment

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

1. The graph of the logarithmic function f(x)=logb(x)+d is shifted upward if _________.
   a. d=0
   b. b. d < 0
   c. c. d > 0
   d. d. o < d < 1

   
   

2. Which of the following is the graph of = log1/2 x ?
   a. [illustration: increasing curve rising steeply to the right of the y-axis on a grid]
   c. [illustration: increasing curve from the lower part of the y-axis, rising slowly to the right, on a grid]
   b. [illustration: decreasing curve from the upper part of the y-axis falling to the right, on a grid]
   d. [illustration: curve approaching a vertical asymptote on the left of the y-axis, falling steeply below the x-axis, on a grid]

   
   

3. The graph of the logarithmic function f(x)=a logb(x) is _________ if 0 < a < 1.
   a. compresses
   b. shifted downward
   c. shifted to the right
   d. stretches

   
   

4. The graph of the logarithmic function f(x)=logb(x+c) is shifted to the right if ________.
   a. c > 0
   b. c < 0
   c. c = 0
   d. 0 < c < 1

   
   

5. Which of the following is the graph of y=log (x+2)?
   a. [illustration: decreasing curve with vertical asymptote on the right of the y-axis, on a grid]
   c. [illustration: increasing curve rising from the right of a vertical line, on a grid]
   b. [illustration: increasing curve with vertical asymptote at x = -2, on a grid]
   d. [illustration: increasing curve with vertical asymptote at x = -2, flatter, on a grid]

   
   

6. Which of the following is the exponential form of −4 = log1/2 x ?
   a. x = −4^(1/2)
   b. −4 = x^(1/2)
   c. x = (1/2)^(−4)
   d. 1/2 = x^(−4)

   
   

7. The graph of the logarithmic function f(x)=logbx is decreasing if_____.
   a. b < 1
   b. b. b = 0
   c. c. 0 < b < 1
   d. d. b > 1

   
   

8. The graph of the logarithmic function f(x)=logb(x)+d is shifted downward if _________.
   a. d=0
   b. d < 0
   c. d > 0
   d. o < d < 1

   
   

9. The graph of the logarithmic function f(x)=logb(x+c) is shifted to the left if ________.
   a. c > 0
   b. b. c < 0
   c. c = 0
   d. 0 < c < 1

   
   

10. The graph of the logarithmic function f(x)=a logb(x) is _________ if 0 a > 1.
    a. compresses
    b. shifted downward
    c. shifted to the right
    d. stretches

    
    

[transcriber’s note: the printed condition in item 10 reads “0 a > 1”; a symbol between 0 and a appears to be missing in the source]

11. The graph of the logarithmic function f(x)=logbx is increasing if_____.
    a. b < 1
    b. b = 0
    c. 0 < b < 1
    d. b > 1

    
    

12. Which of the following is the graph of y=-2logx?
    a. [illustration: decreasing curve with vertical asymptote at the y-axis, on a grid]
    c. [illustration: increasing curve rising steeply near a vertical asymptote at x = 0, on a grid]
    b. [illustration: increasing curve with vertical asymptote at x = 1, on a grid]
    d. [illustration: increasing curve with vertical asymptote at x = -2, on a grid]

    
    

13. How many units of the graph of the logarithmic function f(x)=log(x+3) is shifted to the left?
    a. 2
    b. 3
    c. 4
    d. 5

    
    

14. What is the inverse of y = log3x?
    a. y=x³
    b. b. y=3^x
    c. c. 3^y=x
    d. d. x=y³

    
    

15. What is the inverse of the exponential function?
    a. logarithmic
    b. linear
    c. polynomial
    d. quadratic

    
    

Additional Activities

Sketch the graph of the following logarithmic functions. Show your solution and table of values then describe the graph of the function against the function y=logbx. Write your answer on a separate sheet of paper.

1. y = (x + 5)

Table of values:	Solution:	Graph:
		[chart: blank coordinate grid; x-axis ticks -10, -8, -6, -4, -2, 2, 4, 6, 8, 10; y-axis ticks -10, -8, -6, -4, -2, 2, 4, 6, 8, 10]

[transcriber’s note: the printed function is “y = (x + 5)”; the logarithm symbol appears to be missing in the source]

2. y = x

Table of values:	Solution:	Graph:
		[chart: blank coordinate grid; x-axis ticks -10, -8, -6, -4, -2, 2, 4, 6, 8, 10; y-axis ticks -10, -8, -6, -4, -2, 2, 4, 6, 8, 10]

[transcriber’s note: the printed function is “y = x”; the logarithm symbol appears to be missing in the source]

[transcriber’s note: this page is printed upside down; the three columns are transcribed in reading order]

Answer Key

What I Know

1. d
2. b
3. b
4. d
5. d
6. c
7. c
8. b
9. c
10. a
11. c
12. b
13. a
14. c
15. b

What’s More

Activity 1.1

1. D 2. E 3. A 4. B 5. C

Activity 1.2

y	-3	-2	-1	0	1	2	3
x	1/1000	1/100	1/10	1	10	100	1000

[chart: graph; no title printed] Values are not labeled on the chart.

[chart: graph; no title printed] Values are not labeled on the chart.

[transcriber’s note: the table below is read from a rotated scan; the exponents in the x and y rows are uncertain and are marked]

x	-5 [?]	-4 [?]	-3 [?]	-2 [?]	-1 [?]	0 [?]	1 [?]
y	1/32 [?]	1/16 [?]	1/8 [?]	1/4 [?]	1/2 [?]	1 [?]	2 [?]

Assessment

1. c
2. b
3. a
4. d
5. b
6. c
7. a
8. b
9. a
10. d
11. d
12. c
13. b
14. c
15. a

References

Abramson, Jay. “Graphs of Logarithmic Functions”. Last modified January 3, 2021. https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Precalculus_(OpenStax)/04%3A_Exponential_and_Logarithmic_Functions/4.05%3A_Graphs_of_Logarithmic_Functions.

Bellman, Allan. Advanced Algebra Teacher’s Edition. Prentice Hall 2001.

Caringal, Anthony Zeus. Dynamic of Mathematics (Advanced Algebra with Trigonometry and Introduction to Statistics). Bright House Publishing, 2009.

Catalino D. Mijares, College Algebra Revised Edition, National Bookstore, Inc., 1978, 1979, 1984.

Mini Rose C. Lapinid, Olivia N. Buzon and Gladys C. Nivera, Advanced Algebra, Trigonometry and Statistics: Patterns and Practicalities, Salesiana Books by Don Bosco Press, 2007.

Monterey Institute for Technology and Education, “Introduction to Logarithmic Functions”. Accessed May 29, 2020. http://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U18_L2_T1_text_final.html.

Oronce, Orlando. General Mathematics. Quezon City: Rex Book Store 2016.

Verzosa, Debbie Marie B., Paulo Luis Apolinario, Regina M. Tresvalles, Francis Nelson M. Infante, Jose Lorenzo M. Sin, and Len Patrick Dominic C. Garces. General Mathematics Learner’s Material. First Edition. Department of Education. 2016.

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