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Section 4.8 Applications of Logarithmic Functions. Objectives: 1.To apply logarithmic functions to chemistry, physics, and education. 2.To apply exponential growth to compound interest. Seismologists use the Richter scale to measure earthquake intensity. I. M. log. =. I. - PowerPoint PPT Presentation
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Section 4.8
Applications of Logarithmic Functions
Objectives:1. To apply logarithmic functions to
chemistry, physics, andeducation.
2. To apply exponential growth to compound interest.
Seismologists use the Richter scale to measure earthquake intensity.
0IIlogM =
M is the Richter-scale value.I is the intensity of the earthquake.I0 is the standard minimum intensity.
Earthquake Intensity
EXAMPLE 1 An earthquake has an intensity reading that is 107.5 times that of Io (the standard minimum intensity). What is the measurement of this earthquake on the Richter scale?
M = log IIo
M = log 107.5Io
Io
M = log 107.5
= 7.5
In the field of chemistry, the pH of a substance is defined using logarithms.
pH = –log [H+]
[H+] is the hydrogen ion concentration of the substance in moles per liter.
pH Measurement
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.
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.
Forgetting Curves
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)
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≈ 59.63 (avg. 1 year later)
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?
AnswerS(0) = 78 – 15 log (1)
= 78 – 15(0)= 78
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.
AnswerS(60) = 78 – 15 log (61)
≈ 51.2
A(t) = Pert
A is the total amountr is the annual interest ratet is the time in years
Continuously Compounding Interest
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
= $4971.44
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
= e0.055t800430
ln 1.86 = 0.055t
ln = ln e0.055t800430
= tln 1.860.055
t ≈ 11.3
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)
= $1164.35
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 = 0.0275tt ≈ 25.2
Homework
pp. 213-215
►A. ExercisesFind the Richter-scale measurement for an earthquake that is the given number of times greater than the standard minimum intensity.
1. 106
►A. ExercisesThe 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?
►A. ExercisesThe 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?
►A. ExercisesFind the pH in the substances below according to their given hydrogen ion concentration.
9. Vinegar: [H+] = 7.94 10-4 moles per liter.
►A. ExercisesFind the hydrogen ion concentration (in moles per liter) of the following substances, given their pH values.
11. Hominy: pH = 7.3
►B. ExercisesFind the maximum amount that a person could hope to accumulate from an initial investment of $1000 at
13. 5% interest for 20 years
►B. Exercises17. 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?
►B. Exercises19. How much money was originally
invested in an account if the account totals $51,539.44 after 25 years and interest was compounded continuously at a rate of 6%?
■ Cumulative ReviewFind the domain of each function.
31. p(x) = x2 – 5
■ Cumulative ReviewFind the domain of each function.
32. f(x) = tan x
■ Cumulative ReviewFind the domain of each function. 33. g(x) = 2x + 1
x – 3
■ Cumulative ReviewFind the domain of each function.
34. h(x) = ln x
■ Cumulative ReviewFind the domain of each function. 35. k(x) = x + 2