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