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
M = log
I0
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 107.5 times that of
Io (the standard minimum intensity). What
is the measurement of this earthquake on
the Richter scale?
I
M = log 107.5
M = log
Io
= 7.5
107.5Io
M = log
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
ln 1.86 = 0.055t
800 = 430e0.055t
ln 1.86
=
t
800
0.055
0.055t
=
e
430
t ≈ 11.3
800
ln 430 = ln e0.055t
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
■ Cumulative Review
Find the domain of each function.
2x + 1
33. g(x) =
x–3
■ 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