exponent function

1. 11
General
Mathematics
Quarter 1 – Module 7:
Exponential Functions

2. General Mathematics – Grade 11
Self-Learning Module (SLM)
Quarter 1 – Module 7: Exponential Functions
First Edition, 2020
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Elizabeth G. Torres – EPS, LRMS
Judith B. Alba – EPS, ADM
Reynaldo C. Tagala – EPS, Mathematics

3. 11
General
Mathematics
Quarter 1 – Module 4:
Exponential Functions

4. 4
Introductory Message
For the facilitator:
Welcome to the Grade 11 General Mathematics Self-Learning Module (SLM) on
Exponential 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.

5. 5
For the learner:
Welcome to the Grade 11 General Mathematics Self-Learning Module (SLM) on
Exponential 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 skill
into real life situations or concerns.

6. 6
Assessment This is a task which aims to evaluate your
level of mastery in achieving the learning
competency.
Additional Activities In this portion, another activity will be given
to you to enrich your knowledge or skill of the
lesson learned. This also tends retention of
learned concepts.
Answer Key This contains answers to all activities in the
module.
At the end of this module you will also find:
The following are some reminders in using this module:
1. Use the module with care. Do not put unnecessary mark/s on any part of the
module. Use a separate sheet of paper in answering the exercises.
2. Don’t forget to answer What I Know before moving on to the other activities
included in the module.
3. Read the instruction carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are through with it.
If you encounter any difficulty in answering the tasks in this module, do not
hesitate to consult your teacher or facilitator. Always bear in mind that you are
not alone.
We hope that through this material, you will experience meaningful learning and
gain deep understanding of the relevant competencies. You can do it!
References This is a list of all sources used in developing
this module.

7. 7
What I Need to Know
This module was designed and written with you in mind. It is here to help you
master illustrating Exponential 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.
This module presents the lesson in the following outline:
1. Review: Properties of Exponents
2. Solving exponential equations
3. Solving exponential inequalities
4. Solving problems involving exponential functions
After going through this module, you are expected to:
1. solve exponential equations;
2. solve exponential inequalities; and
3. solve problems involving exponential functions

8. 8
What I Know
Find how much you already know about the topic in this module. Take note of the
items that you were not able to answer correctly and find the right answer as you go
through this module.
A. Determine whether each of the following expression is an exponential equation,
exponential inequality or none of these.
1. 𝑓(𝑥) = 2𝑥3
2. 𝑔(𝑥) = 2𝑥
3. 𝑦 = ⅇ𝑥
4. 22(5𝑥+1) = 500
5. 625 ≥ 5𝑥+8
B. Match Column A with Column B.
Column A Column B
6. 3𝑥
= 81 A. {𝑥 ∈ ℝ|𝑥 ≤ −2}
7. 5𝑥+7
= 125 B. {𝑥 ∈ ℝ|𝑥 < −2}
8. (
4
6
)
𝑥
≥
36
16
C. 𝑥 = 4
9. 5𝑥
> 25𝑥+1
D. 𝑥 = −4
C. Read and solve the problem below.
10. Coffee contains caffeine. The half-life of caffeine is 5 hours . This means
the amount of caffeine in your bloodstream is reduced by 50% every 5
hours. Suppose you drink a cup of coffee that contains 320 mg of caffeine.
How long will it take until there is 5 mg of caffeine left in your bloodstream?

9. 9
Lesson
7
Solving Exponential
Equations, Inequalities, and
Functions
For this module, you need to review your lessons on the definition of zero and
negative exponents and on the laws of exponent.
Definition of Zero and Negative Exponents. Let 𝑎 ≠ 0. We define the following:
1. 𝑎0
= 1
2. 𝑎−𝑛
=
1
𝑎𝑛
Theorem. Let 𝑟 and 𝑠 be rational numbers. Then
1. 𝑎𝑟
𝑎𝑠
= 𝑎𝑟+𝑠
2.
𝑎𝑟
𝑎𝑠 = 𝑎𝑟−𝑠
3. (𝑎𝑟)𝑠
= 𝑎𝑟𝑠
4. (𝑎𝑏)𝑟
= 𝑎𝑟
𝑏𝑟
5. (
𝑎
𝑏
)
𝑟
=
𝑎𝑟
𝑏𝑟
What’s New
Activity 1 Classify Me!
Given:
(a) 49 = 7𝑥+1
(e) 8𝑥 = 𝑥2
− 9
(b) 7 = 2𝑥 + 3 (f) 𝑥2
= 3𝑥3
+ 2𝑥 + 1
(c) 3𝑥
= 32𝑥−1
(g) 2𝑥 + 3 > 𝑥 − 1
(d) 5𝑥−1
= 125 (h) 2𝑥−2
> 8
Questions
1. Which of the given above are exponential equations?
2. Which of the given above are exponential inequalities?
What’s In

10. 10
Activity 2 We have something in common
Complete the table below. Identify the base that can be used in expressing the
given pair of numbers in exponential form. The example is done for you.
Example.
First Number Exponential
Form
Second
Number
Exponential
Form
Common
Base
1. 125 53 25 52 5
First Number Exponential
Form
Second
Number
Exponential
Form
Common
Base
1. 8 32
2 36 216
3 1
4
1
32
4 0.0625 0.5
5 1
100
1
10
What is It
Solving Exponential Equation
One – to – one Property of Exponential Functions will be used in solving
exponential equations.
One to one Property of Exponential Functions
If 𝑥1 ≠ 𝑥2, then 𝑏𝑥1 ≠ 𝑏𝑥2. Conversely, if 𝑏𝑥1 = 𝑏𝑥2then 𝑥1 = 𝑥2
Example 1. Solve the equation 4𝑥−1
= 16
Solution. Write both sides with 4 as the base.
4𝑥−1
= 42
𝑥 − 1 = 2
𝑥 = 2 + 1
𝑥 = 3

11. 11
Alternate Solution. Write both sides with 2 as the base
(22)𝑥−1
= 24
22(𝑥−1)
= 24
2(𝑥 − 1) = 4
2𝑥 − 2 = 4
2𝑥 = 4 + 2
2𝑥 = 6
2𝑥
2
=
6
2
𝑥 = 3
Example 2. Solve the equation 125𝑥−1
= 25𝑥+3
Solution. Both 125 and 25 can be written using 5 as the base.
(53)𝑥−1
= (52)𝑥+3
53(𝑥−1)
= 52(𝑥+3)
3(𝑥 − 1) = 2(𝑥 + 3)
3𝑥 − 3 = 2𝑥 + 6
3𝑥 − 2𝑥 = 6 + 3
𝑥 = 9
Example 3. Solve 9𝑥2
= 3𝑥+3
.
Solution. Both 9 and 3 can be written using 3 as the base.
(32)𝑥2
= 3𝑥+3
32𝑥2
= 3𝑥+3
2𝑥2
= 𝑥 + 3
2𝑥2
− 𝑥 − 3 = 0
(2𝑥 − 3)(𝑥 + 1) = 0
2𝑥 − 3 = 0 or 𝑥 + 1 = 0
2𝑥 = 3 𝑥 = −1
2𝑥
2
=
3
2
𝑥 =
3
2

12. 12
Solving Exponential Inequality
Property of Exponential Inequalities
If 𝑏 > 1, then the exponential function 𝑦 = 𝑏𝑥
is increasing for all x. This means
that 𝑏𝑥
< 𝑏𝑦
if and only if 𝑥 < 𝑦.
If 0 < 𝑏 < 1, then the exponential function 𝑦 = 𝑏𝑥
is decreasing for all x. This
means that 𝑏𝑥
> 𝑏𝑦
if and only if 𝑥 < 𝑦.
Example 4. Solve the inequality 3𝑥
< 9𝑥−2
Solution. Both 9 and 3 can be written using 3 as the base.
3𝑥
< 9𝑥−2
3𝑥
< (32)𝑥−2
3𝑥
< 32(𝑥−2)
3𝑥
< 32𝑥−4
Since the base 3 > 1, then the inequality is equivalent to
𝑥 < 2𝑥 − 4 (the direction of inequality is retained)
4 < 2𝑥 − 𝑥
4 < 𝑥
The solution set to the inequality is {𝑥 ∈ ℝ|𝑥 > 4}
Example 5. Solve the inequality (
1
10
)
𝑥+5
≥ (
1
100
)
3𝑥
Solution. Since,
1
100
= (
1
10
)
2
, then we write both sides of the inequality with
1
10
as
the base.
(
1
10
)
𝑥+5
≥ (
1
100
)
3𝑥
(
1
10
)
𝑥+5
≥ (
1
102)
3𝑥
(
1
10
)
𝑥+5
≥ (
1
10
)
2(3𝑥)
(
1
10
)
𝑥+5
≥ (
1
10
)
6x)

13. 13
Since the base
1
10
< 1, then this inequality is equivalent to
𝑥 + 5 ≤ 6𝑥 (the direction of the inequality is reversed)
5 ≤ 6𝑥 − 𝑥
5 ≤ 5𝑥
5x
5
≤
5
5
1 ≤ 𝑥
The solution set to the inequality is {𝑥 ∈ ℝ|𝑥 ≥ 1}
Solving Problems Involving Exponential Function (Half – Life)
Example 6. The half life of Zn -71 is 2.45 minutes. At 𝑡 = 0 there were 𝑦0 grams of
Zn-71, but only
1
256
of this amount remains after some time. How much time has
passed?
Solution. The half – life of substance can be calculated by the formula 𝑁(𝑡) = 𝑁0 (
1
2
)
𝑡
𝑡
1
2
where 𝑁(𝑡) =quantity of the substance remaining
𝑁0 = initial quantity of the substance
𝑡 = time elapsed
𝑡
1
2
= half life of the substance
Given 𝑡
1
2
= 45 minutes 𝑁0 = 𝑦0
𝑁(𝑡) =
1
256
𝑦0 𝑡=?
Substitute
𝑁(𝑡) = 𝑁0 (
1
2
)
𝑡
𝑡
1
2
1
256
𝑦0 = 𝑦0 (
1
2
)
𝑡
2.45
1
256
= (
1
2
)
𝑡
2.45
(
1
2
)
8
= (
1
2
)
𝑡
2.45
8 =
𝑡
2.45
𝑡 = (8)(2.45)
𝑡 = 19.6 minutes

14. 14
What’s More
Activity 3 What is that animal?
Solve the following exponential equations and inequalities. Write the letter
code on the box that corresponds to the answer of the given expressions.
What is the only animal that never gets sick?
A
162𝑥−3
= 4𝑥+2
H
7𝑥+4
= 492𝑥−1
E
4𝑥+2
= 82𝑥
K
42𝑥+7
≤ 322𝑥−3
R
(
2
5
)
5𝑥−1
≥
25
4
S
(
2
3
)
5𝑥+2
< (
3
2
)
2𝑥
𝑥 >
−2
7
𝑥 = 2
𝑥 =
8
3
𝑥 ≥
−1
5
𝑥 ≥
29
6

15. 15
What I Have Learned
Activity 4 Am I the truth?
Write TRUE if the given statement is correct and write FALSE if the given
statement is wrong.
1. In solving the exponential equation 42𝑥+7
= 322𝑥−3
, 4 and 32 can be
written using 2 as the base.
2. In solving the exponential inequality (0.6)𝑥−3
> (0.6)−2𝑥−2
, the inequality
symbol will not be reversed because 0.6 > 1.
3. The expression 5𝑥−2
> 125 is an example of exponential inequality.
4. The solution of the exponential equation 7𝑥+4
= 492𝑥−1
is 𝑥 = 4.
5. The solution set to the inequality 10𝑥
> 100−2𝑥−5
is {𝑥 ∈ ℝ|𝑥 > −2}
What I Can Do
Activity 5 Solve Me
Solve the following.
1. 162𝑥−3
= 4𝑥+2
2. 42𝑥+7
≤ 322𝑥−3
3. Paracetamol is a drug is used to treat mild to moderate pain and to reduce
fever. It has a half – life in the body of 3 hours. Under a Doctor's care,
Patient N is given a dose of a certain paracetamol. Patient N asks the
doctor for another the same dose of this paracetamol. The doctor agrees to
allow patient N to take dose once there is only 6.25% of the original dose
remaining in the body. How long will patient N have to wait, before taking
another dose of this paracetamol?

16. 16
Assessment
A. Solve for 𝑥.
1. 53𝑥+8
= 252𝑥
2. (
3
5
)
𝑥+1
=
25
9
3. 43𝑥+2
< 64
4. (
49
81
)
𝑥+1
≥
9
7
B. Read and solve the given problem.
5. A certain pain reliever has a half-life of 16 hours. If the initial plasma
level of this pain relieverg, given as a single dose, is 512mg/L, how long
will it take for the plasma level to fall to 16 mg/L?
Additional Activities
Activity 6 Who is right?
1. John and Peter are solving (0.6)𝑥−3
> (0.36)−x−1
. Did anyone get the correct
solution? If not, spot the error or errors.
John Peter
(0.6)𝑥−3
> (0.36)−𝑥−1 (0.6)𝑥−3
> (0.36)−𝑥−1
(0.6)𝑥−3
> (0.62)−𝑥−1 (0.6)𝑥−3
> (0.62)−𝑥−1
(0.6)𝑥−3
> (0.6)2(−𝑥−1) (0.6)𝑥−3
> (0.6)2(−𝑥−1)
(0.6)𝑥−3
> (0.6)−2𝑥−2 (0.6)𝑥−3
> (0.6)−2𝑥−2
𝑥 − 3 > −2𝑥 − 2 𝑥 − 3 < −2𝑥 − 2
𝑥 + 2𝑥 > −2 + 3 𝑥 + 2𝑥 < −2 + 3
3𝑥 > 1 3𝑥 < 1
3𝑥
3
>
1
3
3𝑥
3
<
1
3
𝑥 >
1
3
𝑥 <
1
3

17. 17
Answer Key
What
I
can
do
1.
𝑥
=
8
3
2.
𝑥
≥
29
6
3.
t
=
12
hours
Assessment
1.
𝑥
=
8
2.
𝑥
=
−3
3.
𝑥
<
1
3
4.
𝑥
≤
−
3
2
5.
t
=
80
hours
Additional
Activities
1.
Peter
is
right.
The
inequality
symbol
should
reversed
since
0
⋅
6
<
1
What's
New
1.
(a),
(c),
(d)
2.
(h)
1.
2
3
,
2
5
,
2
2.
6
2
,
6
3
,
6
3.
(
1
2
)
2
,
(
1
2
)
5
,
1
2
4.
(0.5)
2
,
(0.5)
1
,
0.5
5.
(
1
10
)
2
,
(
1
10
)
1
,
1
10
What’s
More
A.
𝑥
=
8
3
H.
𝑥
=
2
E.
𝑥
=
1
K.
𝑥
≥
29
6
R.
𝑥
≥
−1
5
S.
𝑥
>
−2
7
SHARK
What
I
Have
Learned?
1.
TRUE
2.
FALSE
3.
TRUE
4.
FALSE
5.
TRUE
What
I
Know
1.
None
of
these
2.
None
of
these
3.
None
of
these
4.
Exponential
equation
5.
Exponential
inequality
6.
C
7.
D
8.
A
9.
B
10.
30
hours

18. 18
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

19. 19
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.
For inquiries or feedback, please write or call:
Department of Education – SOCCSKSARGEN
Learning Resource Management System (LRMS)
Regional Center, Brgy. Carpenter Hill, City of Koronadal
Telefax No.: (083) 2288825/ (083) 2281893
Email Address: region12@deped.gov.ph