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Section 4.8 Applications of Logarithmic Functions

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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

0

Earthquake Intensity

M is the Richter-scale value.

I is the intensity of the earthquake.

I0 is the standard minimum intensity.

M = log

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 Measurement

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.

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 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

800

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

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

2x + 1

x – 3

■ 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