1. Exponential and Logarithmic Models
Section 3.5
Precalculus PreAP/Dual, Revised ©2018
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§3.5: Exponential and Logarithmic Models

2. Compound Interest Equation
𝑨 = Total Amount Earned
𝑷 = Principle
𝒓 = Interest Rate
𝒏 = Compounded Amount
𝒕 = Time
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§3.5: Exponential and Logarithmic Models

3. §3.5: Exponential and Logarithmic Models
Video
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§3.5: Exponential and Logarithmic Models

4. Compounded Time Frames
Annually: 1 time a year
Semi-Annually: 2 times a year
Quarterly: 4 times a year (not THREE TIMES a year)
Monthly: 12 times a year
Daily: 365 times a year
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§3.5: Exponential and Logarithmic Models

5. §3.5: Exponential and Logarithmic Models
Example 1
$𝟓,𝟎𝟎𝟎 is deposited in an account that pays 𝟔% annual interest compounded quarterly. Find the balance after 25 years.
𝑨= ? How much it is when the balance after 25 years?
𝑷=$𝟓,𝟎𝟎𝟎 $5,000 is deposited
𝒓=𝟎.𝟎𝟔 Interest Rate – remember it needs to be in decimal form
𝒏=𝟒 Compounded quarterly
𝒕=𝟐𝟓 Time it takes to accrue amount
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§3.5: Exponential and Logarithmic Models

6. §3.5: Exponential and Logarithmic Models
Example 1
$𝟓,𝟎𝟎𝟎 is deposited in an account that pays 𝟔% annual interest compounded quarterly. Find the balance after 25 years.
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§3.5: Exponential and Logarithmic Models

7. §3.5: Exponential and Logarithmic Models
Example 2
Determine the amount that a $5,000 investment over ten years at an annual interest rate of 4.8% is worth compounded daily.
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§3.5: Exponential and Logarithmic Models

8. §3.5: Exponential and Logarithmic Models
Example 3
How much must you deposit in an account that pays 6.5% interest, compounded quarterly, to have a balance of $5,000 in 15 years?
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§3.5: Exponential and Logarithmic Models

9. §3.5: Exponential and Logarithmic Models
Your Turn
A deposit is made for $100,000 into an account that pays 6% interest. Find the balance after 10 years if the interest is compounded monthly.
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§3.5: Exponential and Logarithmic Models

10. Compounded Continuously
𝑨 = Total Amount Earned 𝑷 = Principle 𝒆 = The Natural Base 𝒓 = Interest Rate 𝒕 = Time
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§3.5: Exponential and Logarithmic Models

11. §3.5: Exponential and Logarithmic Models
Example 4
A deposit is made for $100,000 into an account that pays 6% interest. Find the balance after 10 years if the interest is compounded continuously.
𝑨 = ??
𝑷 = $100,000
𝒆 = Use e in Calc
𝒓 = 0.06
𝒕 = 10
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§3.5: Exponential and Logarithmic Models

12. §3.5: Exponential and Logarithmic Models
Example 5
An investment of $3,500 at 3% annual interest compounded continuously was made. How much is in the account after 4 years?
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§3.5: Exponential and Logarithmic Models

13. §3.5: Exponential and Logarithmic Models
Your Turn
Suppose that you put in $1,000 into a savings account that compounded continuously. Determine the amount with an interest rate of 5.1% after 10 years.
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§3.5: Exponential and Logarithmic Models

14. §3.5: Exponential and Logarithmic Models
Example 6
You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double?
Doubled Amount
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§3.5: Exponential and Logarithmic Models

15. §3.5: Exponential and Logarithmic Models
Example 6
You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double?
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§3.5: Exponential and Logarithmic Models

16. §3.5: Exponential and Logarithmic Models
Your Turn
How long will it take $30,000 to accumulate to $110,000 in a trust that earns a 10% annual return compounded continuously?
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§3.5: Exponential and Logarithmic Models

17. Exponential Growth/Decay
𝑷 = Ending Amount 𝑷 𝟎 = Initial Amount 𝒆 = The Natural Base 𝒌 = Growth or Decay Rate 𝒕 = Time
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§3.5: Exponential and Logarithmic Models

18. §3.5: Exponential and Logarithmic Models
Example 7
A certain bacterium has an exponential growth rate of 25% per day. If we start with 0.5 grams and provide unlimited resources how much bacteria can we grow in 14 days?
𝑷 = ??
𝑷𝟎 = 0.5
𝒆 = The Natural Base
𝒌 = 0.25
𝒕 = 14 days
P = Ending Amount
P0 = Initial Amount
e = The Natural Base
k = Growth or Decay Rate
t = Time
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§3.5: Exponential and Logarithmic Models

19. §3.5: Exponential and Logarithmic Models
Example 8
What is the total amount of bacteria when the initial amount of bacteria is 300, 𝒌=𝟎.𝟎𝟔𝟖, and the time studied is 52 hours?
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§3.5: Exponential and Logarithmic Models

20. §3.5: Exponential and Logarithmic Models
Your Turn
At the start of an experiment, there are 100 bacteria. If the bacteria follow an exponential growth pattern with rate 𝒌=𝟎.𝟎𝟐, what will be the population after 5 hours?
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§3.5: Exponential and Logarithmic Models

21. §3.5: Exponential and Logarithmic Models
Example 9
During its exponential growth phase, a certain bacterium can grow from 5,000 cells to 12,000 cells in 10 hours. What is the growth rate?
P = 12,000
P0 = 5,000
𝒆 = The Natural Base
k = ??
t = 10
P = Ending Amount
P0= Initial Amount
e = The Natural Base
k = Growth or Decay Rate
t = Time
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§3.5: Exponential and Logarithmic Models

22. §3.5: Exponential and Logarithmic Models
Example 9
During its exponential growth phase, a certain bacterium can grow from 5,000 cells to 12,000 cells in 10 hours. What is the growth rate?
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§3.5: Exponential and Logarithmic Models

23. §3.5: Exponential and Logarithmic Models
Example 10
During its exponential growth phase, a certain bacterium can grow from 5,000 cells to 15,000 cells in 12 hours. What is the growth rate?
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§3.5: Exponential and Logarithmic Models

24. §3.5: Exponential and Logarithmic Models
Your Turn
The population of a certain city in 2000 was 99,500. What is its initial population in 1975 when its growth rate is at Round to the nearest whole number.
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§3.5: Exponential and Logarithmic Models

25. §3.5: Exponential and Logarithmic Models
Example 11
If certain isotope has a half-life of 4.2 days. How long will it take for a 150 milligram sample to decay so that only 10 milligrams are left?
𝑷 = 75
𝑷𝟎 = 150
𝒆 = The Natural Base
𝒌 = ??
𝒕 = 4.2
P = Ending Amount
P0= Initial Amount
e = The Natural Base
k = Growth or Decay Rate
t = Time
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§3.5: Exponential and Logarithmic Models

26. §3.5: Exponential and Logarithmic Models
Example 11
If certain isotope has a half-life of 4.2 days. How long will it take for a 150 milligram sample to decay so that only 10 milligrams are left?
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§3.5: Exponential and Logarithmic Models

27. §3.5: Exponential and Logarithmic Models
Example 11
If certain isotope has a half-life of 4.2 days. How long will it take for a 150 milligram sample to decay so that only 10 milligrams are left?
𝑷 = 10
𝑷𝟎 = 150
𝒆 = The Natural Base
𝒌 = –.1650
𝒕 = ??
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§3.5: Exponential and Logarithmic Models

28. §3.5: Exponential and Logarithmic Models
Example 12
The half-life of carbon-14 is 5,730 years. The skeleton of a mastodon has 42% of its original Carbon-14. When did the mastodon die?
𝑷 = ½ (half life)
𝑷𝟎 = 1 (full life)
𝒆 = The Natural Base
𝒌 = ??
𝒕 = 5,730
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§3.5: Exponential and Logarithmic Models

29. §3.5: Exponential and Logarithmic Models
Example 12
The half-life of carbon-14 is 5,730 years. The skeleton of a mastodon has 42% of its original Carbon-14. When did the mastodon die?
𝑷 = 0.42 (total left)
𝑷𝟎 = 1
𝒆 = The Natural Base
𝒌 = (ln 0.5)/5730
𝒕 = ??
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§3.5: Exponential and Logarithmic Models

30. §3.5: Exponential and Logarithmic Models
Your Turn
The half-life of carbon-14 is 5,730 years. If it is determined that an old bone contains 𝟖𝟓% of its original carbon-14, how old is the bone?
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§3.5: Exponential and Logarithmic Models

31. Newton’s Law of Cooling
𝑻 𝑭 = Final Temperature 𝑻 𝑹 = Temperature of the Environment 𝑻 𝟎 = Initial Temperature of the Object 𝒆 = The Natural Base 𝒌 = Growth or Decay Rate 𝒕 = Time
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§3.5: Exponential and Logarithmic Models

32. §3.5: Exponential and Logarithmic Models
Example 13
A container of ice cream arrives home from the supermarket at a temperature of 𝟔𝟓°𝑭. It is placed in the freezer which has a temperature of 𝟐𝟎°𝑭. Determine the final temperature at which it will be still considered “freezing,” if the rate of change is 𝟎.𝟏𝟎𝟕°𝑭 per minute for 𝟏𝟐.𝟑𝟓 minutes.
TF = Final Temperature
TR = Environment Temp
T0 = Initial Temperature
e = The Natural Base
k = Growth or Decay Rate
t = Time
𝑻𝑭 = ??
𝑻𝑹 = 20°
𝑻𝟎 = 65°
𝒆 = The Natural Base
𝒌 = 0.107
𝒕 = 12.35
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§3.5: Exponential and Logarithmic Models

33. §3.5: Exponential and Logarithmic Models
Your Turn
The cooling model for tea served in a 6 oz. cup uses Newton’s Law of Cooling equation. The original temperature was 𝟐𝟎𝟎°𝑭 and current environment temperature of the tea is at 𝟔𝟖°𝑭. Determine the temperature if the decay rate is at 𝟎.𝟒𝟏 per minute and waiting time is 6 minutes.
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§3.5: Exponential and Logarithmic Models

34. §3.5: Exponential and Logarithmic Models
Example 14
When an object is removed from a furnace and placed in an environment with a constant decay rate of 𝟎.𝟑𝟏𝟏𝟒 and the room temperature of 𝟖𝟎°𝑭, its core temperature is 𝟏𝟓𝟎𝟎°𝑭. If the final temperature is at 𝟑𝟕𝟖°𝑭, about how long is it out of the furnace (in hours)?
TF = Final Temperature
TR = Environment Temp
T0 = Initial Temperature
e = The Natural Base
k = Growth or Decay Rate
t = Time
𝑻𝑭 = 𝟑𝟕𝟖°
𝑻𝑹 = 𝟖𝟎°
𝑻𝟎 = 𝟏𝟓𝟎𝟎°
𝒆 = The Natural Base
𝒌 = 𝟎.𝟑𝟏𝟏𝟒
𝒕 = ??
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§3.5: Exponential and Logarithmic Models

35. §3.5: Exponential and Logarithmic Models
Example 14
When an object is removed from a furnace and placed in an environment with a constant decay rate of 𝟎.𝟑𝟏𝟏𝟒 and the room temperature of 𝟖𝟎°𝑭, its core temperature is 𝟏𝟓𝟎𝟎°𝑭. If the final temperature is at 𝟑𝟕𝟖°𝑭, about how long is it out of the furnace (in hours)?
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§3.5: Exponential and Logarithmic Models

36. §3.5: Exponential and Logarithmic Models
Example 14
Pete was driving on a hot day when the car starts overheating and stops running. It overheats to 𝟐𝟖𝟎°𝑭 and can be driven again at 𝟐𝟑𝟎°𝑭. Suppose it takes 𝟔𝟎 minutes until Pete can drive if is 𝟖𝟎°𝑭 outside, what is the decay factor? Round to three decimal places.
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§3.5: Exponential and Logarithmic Models

37. §3.5: Exponential and Logarithmic Models
Your Turn
Devin baked a yam at 𝟑𝟓𝟎°, and when Devin removed it from the oven, he let the yam cool, which has a room temperature of 𝟔𝟖°𝑭. After 𝟏𝟎 minutes, the yam has cooled to 𝟐𝟒𝟎°𝑭. What is the decay factor?
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§3.5: Exponential and Logarithmic Models

38. §3.5: Exponential and Logarithmic Models
Assignment
Worksheet
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§3.5: Exponential and Logarithmic Models