Applications of Logarithmic Functions 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.

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

EXAMPLE 1 An earthquake has an intensity reading that is 107 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 I Io M = log 107.5 = 7.5 M = log 107.5Io Io

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

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

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.

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.

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? Answer S(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. Answer S(60) = 78 – 15 log (61) ≈ 51.2

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

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 ln 1.86 = 0.055t = t ln 1.86 0.055 = e0.055t 800 430 t ≈ 11.3 ln = ln e0.055t 800 430

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 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 = 0.0275t t ≈ 25.2

Homework pp. 213-215

►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

►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?

►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?

►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.

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

►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

►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?

►B. Exercises 19. 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 Review Find the domain of each function. 31. p(x) = x2 – 5

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

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

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

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