1. Applications of Logarithmic Functions
Section 4.8
Applications of Logarithmic Functions

2. Objectives:
1. To apply logarithmic functions to
chemistry, physics, and
education.
2. To apply exponential growth to
compound interest.

3. Seismologists use the Richter scale to measure earthquake intensity.

4. Earthquake Intensity I
log M =
M is the Richter-scale value.
I is the intensity of the earthquake.
I0 is the standard minimum intensity.

5. EXAMPLE 1 An earthquake has an intensity reading that is 107
EXAMPLE 1 An earthquake has an intensity reading that is times that of Io (the standard minimum intensity). What is the measurement of this earthquake on the Richter scale?
M = log I Io
M = log 107.5
= 7.5
M = log
107.5Io Io

6. In the field of chemistry, the pH of a substance is defined using logarithms.

7. pH Measurement
pH = –log [H+]
[H+] is the hydrogen ion concentration of the substance in moles per liter.

8. EXAMPLE 2 Determine the pH of milk if the hydrogen ion concentration is 4  10-7 moles per liter.
pH = -log [H+]
pH = -log [4  10-7]
pH = -[log 4 + log 10-7]
= -[log 4 + (-7)]
≈ 6.4
The pH of milk is 6.4.

9. Forgetting Curves
The equation for the average test score on previously learned material.
S(t) = A - B log (t + 1).
t is the time in months.
A and B are constants found by experimentation in a course.

10. EXAMPLE 3 If the average score in a geometry class for a certain exam is given by s(t) = 73 – 12 log (t + 1), what was the original average score? What will the average score be on the same exam a year later?
s(t) = 73 – 12 log (t + 1)
s(0) = 73 – 12 log (0 + 1)
= 73 – 12(0)
= 73 (the original average test score)

11. EXAMPLE 3 If the average score in a geometry class for a certain exam is given by s(t) = 73 – 12 log (t + 1), what was the original average score? What will the average score be on the same exam a year later?
s(t) = 73 – 12 log (t + 1)
s(12) = 73 – 12 log (12 + 1)
= 73 – 12 log 13
≈ (avg. 1 year later)

12. Practice: If the average score in a geometry class is given by S(t) = 78 – 15 log (t + 1), what was the original average score?
Answer
S(0) = 78 – 15 log (1)
= 78 – 15(0)
= 78

13. Practice: If the average score in a geometry class is given by S(t) = 78 – 15 log (t + 1), what would the average score be after 5 years? Round to the nearest tenth.
Answer
S(60) = 78 – 15 log (61)
≈ 51.2

14. Continuously Compounding Interest
A(t) = Pert
A is the total amount
r is the annual interest rate
t is the time in years

15. EXAMPLE 4 $400 is deposited in a savings account with an interest rate of 6% for a period of 42 years. How much money will be in the account at the end of 42 years if interest is compounded continuously?
A(t) = Pert
A(42) = 400e(0.06)(42)
= 400e2.52
= $

16. EXAMPLE 5 How long will it take Shannon to save $800 from an initial investment of $430 at 5½% interest with continuous compounding?
A(t) = Pert
800 = 430e0.055t
ln 1.86 = 0.055t
= t
ln 1.86
0.055
= e0.055t
800
430
t ≈ 11.3
ln = ln e0.055t
800
430

17. Practice: $550 is deposited in a savings account with an interest rate of 5%. How much money will be in the account after 15 years if interest is compounded continuously?
Answer
A(t) = 550e(0.05)(15)
= $

18. Practice: How long will it take $800 to double at 2
Practice: How long will it take $800 to double at 2.75% interest with continuous compounding? Round to the nearest tenth.
Answer
1600 = 800e0.0275t
2 = e0.0275t
ln 2 = t
t ≈ 25.2

19. Homework pp

20. ►A. Exercises
Find the Richter-scale measurement for an earthquake that is the given number of times greater than the standard minimum intensity.
1. 106

21. ►A. Exercises
The formula for the average score on a particular English exam after t months is S(t) = 82 – 8 log (t + 1).
5. What is the average score after 5 months?

22. ►A. Exercises
The formula for the average score on a particular English exam after t months is S(t) = 82 – 8 log (t + 1).
7. If a group of people lived for 40 years after taking this English exam and took the test again, what would the average score be?

23. ►A. Exercises
Find the pH in the substances below according to their given hydrogen ion concentration.
9. Vinegar: [H+] = 7.94  10-4 moles per liter.

24. ►A. Exercises
Find the hydrogen ion concentration (in moles per liter) of the following substances, given their pH values.
11. Hominy: pH = 7.3

25. ►B. Exercises
Find the maximum amount that a person could hope to accumulate from an initial investment of $1000 at
13. 5% interest for 20 years

26. ►B. Exercises
17. How much money is in an account after 15 years if the interest is compounded continuously at a rate of 7% and the original principal was $5000?

27. ►B. Exercises
19. How much money was originally invested in an account if the account totals $51, after 25 years and interest was compounded continuously at a rate of 6%?

28. ■ Cumulative Review Find the domain of each function.
31. p(x) = x2 – 5

29. ■ Cumulative Review
Find the domain of each function.
32. f(x) = tan x

30. ■ Cumulative Review Find the domain of each function. 33. g(x) =

31. ■ Cumulative Review
Find the domain of each function.
34. h(x) = ln x

32. ■ Cumulative Review
Find the domain of each function.
35. k(x) = x + 2