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Applications of Logarithmic Functions

1. To apply logarithmic functions to

chemistry, physics, and

education.

2. To apply exponential growth to

compound interest.

M

log

=

I

0

Earthquake Intensity

M is the Richter-scale value.

I is the intensity of the earthquake.

I0 is the standard minimum intensity.

I

Io

107.5Io

Io

M = log

EXAMPLE 1An 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 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.

EXAMPLE 2Determine 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.

EXAMPLE 3If 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 3If 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

430

800

430

= t

ln 1.86

0.055

= e0.055t

ln = ln e0.055t

EXAMPLE 5How 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 ≈ 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 = 0.0275t

t ≈ 25.2

pp. 213-215

Find the Richter-scale measurement for an earthquake that is the given number of times greater than the standard minimum intensity.

1. 106

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?

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?

Find the pH in the substances below according to their given hydrogen ion concentration.

9. Vinegar: [H+] = 7.94 10-4 moles per liter.

Find the hydrogen ion concentration (in moles per liter) of the following substances, given their pH values.

11. Hominy: pH = 7.3

Find the maximum amount that a person could hope to accumulate from an initial investment of $1000 at

13. 5% interest for 20 years

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?

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%?

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