Exponential function

1. Review : Positive, Zero and negative exponents
𝒂𝒏
= 𝒂 ∗ 𝒂 ∗ 𝒂
𝒂𝟎
= 𝟏
𝒂−𝒏
=
𝟏
𝒂𝒏
𝒂𝟐
= 𝒂 ∗ 𝒂
𝟏𝟎𝟎𝟎𝟎
= 𝟏
𝒂−𝟒
=
𝟏
𝒂𝟒
Examples:

2. Review : Law of exponents
𝟏. 𝒂𝒏
𝒂𝒎
= 𝒂𝒏+𝒎
𝟐. (𝒂𝒏
)𝒎
= 𝒂(𝒏)(𝒎)
𝟒.
𝒂
𝒃
𝒏
=
𝒂𝒏
𝒃𝒏
𝟑. (𝒂𝒃)𝒏
= 𝒂𝒏
𝒃𝒏
𝟓.
𝒂𝒏
𝒂𝒎
= 𝒂𝒏−𝒎
Product of powers
Powers of powers
Powers of product rule
Powers of Quotient rule
Quotient rule

3. Review : Law of exponents
𝟏. (𝒙𝟑
) 𝒙𝟒
𝟐. (𝒚𝟑
)𝒙
𝟑. (𝒙𝒚𝒛)𝟒
𝟒.
𝒙
𝒚
−𝟐
𝟓.
𝒚𝟓
𝒚𝟑

4. Contents
Exponential
Function
Logarithmic
Function

6. Exponential
Function

7. Pay it forward
What did you ever do to change the world?

8. Pay it forward

9. ENTER YOUR LEARNING TARGET
● I can represent real-life situation using
exponential function.
● I can define exponential function.

10. Exponential function
An exponential function with base b is a
function of the form
𝑓 𝑥 = 𝑏𝑥
, 𝑤ℎ𝑒𝑟𝑒 𝑏 > 0 𝑎𝑛𝑑 𝑏 ≠ 1

11. base
An exponential function with base b is a function of
the form
𝑓 𝑥 = 𝑏𝑥
, 𝑤ℎ𝑒𝑟𝑒 𝑏 > 0 𝑎𝑛𝑑 𝑏 ≠ 1

12. Examples of Exponential Function
Examples of not a Exponential Function
Note: 𝒃 > 𝟎, 𝒂𝒏𝒅 𝒃 ≠ 𝟏

13. Proof that b>0 Note: 𝒃 > 𝟎, 𝒂𝒏𝒅 𝒃 ≠ 𝟏
Counter example:
𝒃 = −𝟒 𝒂𝒏𝒅 𝒙 =
𝟏
𝟐
𝒇 𝒙 = 𝒃𝒙
𝒇 𝒙 = −𝟒
𝟏
𝟐
𝒇 𝒙 = −𝟒
𝒇 𝒙 = 𝟐𝒊

15. Complete a table of values for x = -3, -2, -1, 0, 1, 2
x -3 -2 -1 0 1 2 3
𝑓 𝑥 =
1
3
𝑥
𝑓 𝑥 = 10𝑥
𝑓 𝑥 = 0.80𝑥

16. Many applications involve transformation of
exponential functions. Some of the most common
applications in real-life exponential functions
and their transformations are population
growth and exponential decay, and compound
interest.
Application of Exponential Function

17. On several instances, scientist
will start with a certain number
of bacteria or animals and
watch how the population grows.
For example, if the population
doubles every 3 days, this can be
represented as an exponential
function.
IDEA

18. Suppose that the quantity of bacteria 𝒚
doubles every 𝑻units of time. If 𝒚𝟎is the
initial amount, then the quantity 𝒚 after t
unit of time is given by 𝒚 = 𝒚𝟎 𝟐 𝒕/𝑻
Situation

19. Let t= time in days. At t=0, there were initial 20 bacteria. Suppose that the
bacteria doubles every 100 hours. Give an exponential model for the bacteria as
function of t.
Problem 1 Population
Initially, at t=0 Number of bacteria = 20
Initially, at t=300 Number of bacteria = 20(2)(2)(2)
Initially, at t=200 Number of bacteria = 20(2)(2)
Initially, at t=100 Number of bacteria = 20(2)

20. At a time t=0, 500 bacteria are in petri dish, and
this amount of bacteria triples every 15 days.
a. Give an exponential model for this situation.
b. How many bacteria are in the petri dish after
40 days.
Problem 2 Population

21. At a time t=0, 500 bacteria are in petri dish, and this amount of
bacteria triples every 15 days.
a. Give an exponential model for this situation.
b. How many bacteria are in the petri dish after 40 days.
Problem 2 Population
Initially, at t=0 Number of bacteria = 500
Initially, at t=30 Number of bacteria = 500(3)(3)
Initially, at t=15 Number of bacteria = 500(3)
𝑓 𝑥 = 500 3
𝑡
15

22. At a time t=0, 500 bacteria are in petri dish, and this amount of bacteria triples
every 15 days.
b. How many bacteria are in the petri dish after 40 days.
Problem 2 Population
𝑓 𝑥 = 500 3
𝑡
15
𝑓 𝑥 = 500 3
40
15
𝑡 = 40
𝑓 𝑥 = (500)
15
340
𝑓 𝑥 = (500)(18.72075441)
𝑓 𝑥 ≈ 9360.38
Therefore, there would be 9360
bacteria in the petri dish within 15
days.

23. Exponential
EQUATION

24. EXPONENTIAL EQUATION
An equation involving exponential expression.
𝑦 = 2𝑥

25. Examples:
3𝑥
+ 3 = 30
52𝑥
= 625

26. ●

27. Write both sides of the
equations as powers of the same
base.
Strategy in solving Exponential
Equations.

28. Exercises:
1. 4𝑥−1
= 16
2. 125𝑥−1
= 25𝑥+3
3. 9𝑥2
= 3𝑥+3
4.
1
125
2𝑥−2
∗ 625𝑥
= 125
5. 4−𝑛−2
∗ 4𝑛
= 64

29. The Base “e” (also called the natural base)
To model things in nature, we’ll
need a base that turns out to be
between 2 and 3. Your calculator
knows this base. Ask your
calculator to find e1. You do this by
using the ex button (generally you’ll
need to hit the 2nd or yellow button
first to get it depending on the
calculator). After hitting the ex, you
then enter the exponent you want
(in this case 1) and push = or enter.
If you have a scientific calculator
that doesn’t graph you may have to
enter the 1 before hitting the ex.
You should get 2.718281828

30. In a certain quantity increases by a fixed
percent each year (or any time of period ,
the amount of that quantity after t years
can be modelled by equation:
𝒚 = 𝒂𝟎 𝟏 + 𝒓 𝒕
Representation

31. 𝒚 = 𝒂𝟎 𝟏 + 𝒓 𝒕
Where a is the initial amount and r is the
percent increased express as decimal . In
this case the quantity 𝟏 + 𝒓 is called
growth factor
Representation

32. 1. A town has a population of 40,000 that is increasing at
the rate of 5% each year. Find the population of the town
after 6 years.
Example1
𝑆𝑡𝑒𝑝 1: 𝑊𝑟𝑖𝑡𝑒 𝑡ℎ𝑒 𝑔𝑖𝑣𝑒𝑛 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛
𝑦 = 𝑦0 1 + 𝑟
𝑡
𝑇
𝑦 =?
𝑦0 = 40000,
𝑟 = 5% 𝑜𝑟 0.05,
𝑡 = 6 𝑦𝑒𝑎𝑟𝑠, 𝑎𝑛𝑑
𝑇 = 1 ( 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑤𝑜𝑟𝑑 𝑒𝑎𝑐ℎ 𝑦𝑒𝑎𝑟)

33. 1. A town has a population of 40,000 that is increasing at the rate of 5%
each year. Find the population of the town after 6 years.
Given:
𝑦 = 𝑦0 1 + 𝑟
𝑡
𝑇
𝑦 = ?
𝑦0 = 40000,
𝑟 = 5% 𝑜𝑟 0.05,
𝑡 = 6 𝑦𝑒𝑎𝑟𝑠, 𝑎𝑛𝑑
𝑇 = 1 ( 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒
𝑤𝑜𝑟𝑑 "𝑒𝑎𝑐ℎ 𝑦𝑒𝑎𝑟")
𝑆𝑡𝑒𝑝 2: 𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑡ℎ𝑒 𝑔𝑖𝑣𝑒𝑛 𝑣𝑎𝑙𝑢𝑒
𝑦 = 𝑦0 1 + 𝑟
𝑡
𝑇
𝑦 = 40,000 1 + 0.05
6
1
𝑆𝑡𝑒𝑝 3 𝐷𝑜 𝑡ℎ𝑒 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛𝑠
𝑦 = 40,000 1.05 6
𝑦 ≈ 53603.8 ≈ 53604
Therefore, the population
after 6 years will be
53604.

34. The population of a province is 75,000, and has
been increasing at the rate of 2.5% a year for the
past 15 years. What was the population 15 years
ago?
Example 2

35. The population of a province is 75,000, and has been increasing at the rate of
2.5% a year for the past 15 years. What was the population 15 years ago?
Example 2
Given:
𝑦 = 𝑦0 1 + 𝑟
𝑡
𝑇
𝑦 = 75000
𝑦0 =?
𝑟 = 2.5% 𝑜𝑟 0.025,
𝑡 = 15 𝑦𝑒𝑎𝑟𝑠, 𝑎𝑛𝑑
𝑇 = 1 ( 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒
𝑤𝑜𝑟𝑑 "𝑎 𝑦𝑒𝑎𝑟")
𝑆𝑡𝑒𝑝 2: 𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑡ℎ𝑒 𝑔𝑖𝑣𝑒𝑛 𝑣𝑎𝑙𝑢𝑒
75,000 = 𝑦0 1 + 0.025
15
1
75,000 = 𝑦0 1.025
15
1
75,000
1.448298116
=
𝑦0 1.448298116
1.448298116
51784.91 ≈ 𝑦0
Therefore, the population
15 years before was
51785.

36. A mine worker discovers an ore sample containing
500 mg of radioactive material. It is discovered
that the radioactive material has a half life of 1
day. Find the amount of radioactive material in the
sample at the beginning of the 7th day?
Example 3

38. A mine worker discovers an ore sample containing 500 mg of radioactive
material. It is discovered that the radioactive material has a half life of 1
day. Find the amount of radioactive material in the sample at the beginning
of the 7th day?
Example 3
Given: 𝐴 = 𝐴0 ∗
1
2
𝑡
ℎ
𝐴0 = 500,
𝑠𝑝𝑙𝑖𝑡 𝑓𝑎𝑐𝑡𝑜𝑟 =
1
2
𝑏𝑒𝑐𝑎𝑢𝑠𝑒 𝑜𝑓 ℎ𝑎𝑙𝑓 − 𝑙𝑖𝑓𝑒
𝑡 = 7
ℎ = 1

39. A mine worker discovers an ore sample containing 500 mg of radioactive
material. It is discovered that the radioactive material has a half life of 1
day. Find the amount of radioactive material in the sample at the beginning
of the 7th day?
Example 3
𝐴 = 500 ∗
1
2
7
1
𝐴 = 500 ∗ (
17
27
)
𝐴 = 500 ∗
1
128
= 3.90625
Given:
𝐴 = 𝐴0 ∗
1
2
𝑡
ℎ
𝐴0 = 500
𝑠𝑝𝑙𝑖𝑡 𝑓𝑎𝑐𝑡𝑜𝑟 =
1
2
𝑡 = 7
ℎ = 1
Therefore, the amount of
radioactive material in
the beginning of the 7th
day was 3.90

40. 𝑨 = 𝑷 𝟏 +
𝒓
𝒏
𝒏𝒕
Exponential on Compound Interest
Where;
𝐴 = 𝐶𝑜𝑚𝑝𝑜𝑢𝑛𝑑 𝐴𝑚𝑜𝑢𝑛𝑡
𝑃 = 𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙
𝑟 = 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑅𝑎𝑡𝑒
𝑡 = 𝑡𝑖𝑚𝑒 𝑖𝑛 𝑦𝑒𝑎𝑟𝑠
𝑛 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑎𝑝𝑝𝑙𝑖𝑒𝑑 𝑝𝑒𝑟 𝑡𝑖𝑚𝑒 𝑝𝑒𝑟𝑖𝑜𝑑

41. Ramon won P35,000 in mathematics quiz bee.
He deposited it in the bank which gave an
annual interest of 6% compounded monthly.
Assuming he did for a period of 10 years, how
much will his money be by then?
Example 4

42. Ramon won P35,000 in mathematics quiz bee. He deposited it in the bank
which gave an annual interest of 6% compounded monthly. Assuming he did for
a period of 10 years, how much will his money be by then?
Example 4
Given:
𝐴 =?
𝑃 = 35,000
𝑟 = 0.06
𝑡 = 10
𝑛 = 1(𝑏𝑒𝑐𝑎𝑢𝑠𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑤𝑜𝑟𝑑 "annual=yearly")
𝑨 = 𝟑𝟓𝟎𝟎𝟎 𝟏 +
𝟎. 𝟎𝟔
𝟏
(𝟏)(𝟏𝟎)
𝑨 = 𝟑𝟓𝟎𝟎𝟎 𝟏. 𝟎𝟔 (𝟏𝟎)
𝑨 = 𝟑𝟓𝟎𝟎𝟎(𝟏. 𝟕𝟗𝟎𝟖𝟒𝟕𝟔𝟗𝟕)
𝑨 ≈ 𝟔𝟐𝟔𝟕𝟗. 𝟔𝟕
Therefore, the money will be
P62,679.67 in 10 years.