The document provides an educational module on solving exponential equations, defining exponential functions, and illustrating real-life applications such as population growth and compound interest. It includes exercises and activities for students to practice simplifications, create exponential models, and solve related problems. An assessment component is also included to evaluate understanding of the material.

1. SOLVING
EXPONENTIAL
EQUATIONS
ANNALYN MANIEGO CABATINGAN

2. WHAT I NEED
TO KNOW
01
After going through this module, you are
expected to:
1. define exponential functions;
2. show illustrations of exponential
functions that represent real-life
situations;
3. represent real-life situations using the
exponential functions; and
4. solve problems involving real-life
situations using the exponential
functions.

3. REVIEW ON
LAWS OF EXPONENT

4. Simplify the following.
ASSIGNMENT NO.1

5. Simplify the
following.
SEATWORK NO. 2

6. SOLVING EXPONENTIAL EQUATIONS
One-to-One Property
If , then .
Conversely, if ,
then .

7. Solve the following.
EXERCISES:

8. EXPONENTIAL FUNCTION
with base b is a function of
the form

9. Example 1:
If , evaluate

10. Example 2: Complete the table of values
for exponential function
𝒚 =𝟑
𝒙
, 𝐚𝐧𝐝 𝒚 =( 𝟏
𝟑 )
𝒙
.
x

11. Population Growth by Doubling
Suppose a quantity y doubles every T units of
time. If is the initial amount, then the quantity y
after t units of time is given by

12. Example 3:
At , there are initially 20
bacteria. Suppose that the
bacteria doubles every 100
hours, give an exponential
model for the bacteria as a
function of

13. Exponential Decay by Half-life
If the half-life of a substance is T units, and is the
amount of the substance corresponding to , then
the amount y of substance remaining after t units
of time is given by

14. Example 4: Suppose that the half-
life of a certain radioactive
substance is 10 days and there
are 10g initially. Determine the
amount of substance remaining
after 30 days and give an
exponential model for the
amount of remaining
substance.

15. Compound Interest
If a principal P is invested at an annual rate of r,
compounded annually, then the amount after t
years is given by

16. Example 5: Mrs. Cruz invested P100,000.00 in a company
that offers 6% interest compounded annually. Define an
exponential model for this situation. How much will this
investment be worth at the end of each year for the next
five years?

17. CREDITS: This presentation template was created by Slidesgo, including
icons by Flaticon, and infographics & images by Freepik
What’s More
Activity 1.1
Solve the following:
1. A culture of 100 bacteria in a petri dish doubles every hour.
a. Complete the table.
b. Write the exponential model for the number of bacteria inside
the box.
t 0 1 2 3 4
No. of
bacter
ia

18. CREDITS: This presentation template was created by Slidesgo, including
icons by Flaticon, and infographics & images by Freepik
What’s More
Activity 1.1
Solve the following:
1. A culture of 100 bacteria in a petri dish doubles every hour.
c. How many bacteria will there be after 6 hours?

19. CREDITS: This presentation template was created by Slidesgo, including
icons by Flaticon, and infographics & images by Freepik
What’s More
Activity 1.1
Solve the following:
2. The half-life of a radioactive substance is 12 hours and there are
100 grams initially.
a. Complete the table.
b. Write the exponential model for the amount of substance
inside the box.
t 0 12 24 36 48
No. of
bacteri
a

20. CREDITS: This presentation template was created by Slidesgo, including
icons by Flaticon, and infographics & images by Freepik
What’s More
Activity 1.1
Solve the following:
2. The half-life of a radioactive substance is 12 hours and there are
100 grams initially.
c. Determine the amount of substance left after 3 days.

21. CREDITS: This presentation template was created by Slidesgo, including
icons by Flaticon, and infographics & images by Freepik
What’s More
Activity 1.1
Solve the following:
3. Kim deposited ₱10,000.00 in a bank that pays a 3% compound
interest annually.
a. Identify the given:
P = _____
r = _____
b. Write the exponential model inside the box.

22. CREDITS: This presentation template was created by Slidesgo, including
icons by Flaticon, and infographics & images by Freepik
What’s More
Activity 1.1
Solve the following:
3. Kim deposited ₱10,000.00 in a bank that pays a 3% compound
interest ann.
c. How much money will he have after 2 years?

23. What I Have Learned
1. A function of the form ( ) = or = , where > 0 and
𝑓 𝑥 𝑏𝑥 𝑦 𝑏𝑥 𝑏 𝑏
≠ 1 is called _____________.
2. Suppose a quantity y doubles every T units of time. If yo is the
initial amount, then the quantity y after t units is given by the
formula __________________.
3. The time it takes for half of the substance to decay is called
_____________.
4. The exponential decay formula is ________________.
5. If a principal P is invested at an annual rate of r; compounded
annually, then the amount after t years is given by the formula
________________.

24. What I Can Do
Read and understand the situation below, then answer the questions or perform
the tasks that follow.
Wise Decision and Friendship Goal
You have a best friend, and she is also an 18-year old senior high school
student and asking for your advice as to which between the two “25th
birthday gift options” posted by her parents she should choose for her
25th birthday.
Option A: Her parents will give her ₱3,000.00 each year starting from her
19th
birthday until her 25th birthday.
Option B: Her parents will give her ₱400.00 on her 19th birthday, ₱800.00
on her 20th birthday, ₱1,600.00 on her next birthday, and the

25. What I Can Do
Task: You need to prepare a written
report highlighting the amount
of money (y) your best friend
gets each year (x) starting from
her 19th birthday using options
A and B in tabular form. Write
equations that represent the
two options with a complete set
of solutions. At the end of your
report, write a conclusion
stating the option you will
choose and the explanation of

26. What I Can Do
Rubrics for rating the output:
SCORE DESCRIPTORS
20
The situation is correctly modeled with an exponential 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 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 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 function, a solution is presented but
has an incorrect final answer.
The additional 5 points will be determined from the conclusions or
justifications made.

27. What I Know
01
02
Activity 1.2
03
Performance Task No. 4
What I Have Learned
What’s More
Answer the following:
What I Can Do
04
05
Assessment

28. If you don’t play by
the rule, don’t
complain if you
lose.