Section 8D
Logarithm Scales: Earthquakes,
Sounds, and Acids
Pages 546-558
Measurement Scales
Earthquake strength is described in magnitude.
Loudness of sounds is described in decibels.
Acidity of solutions is described by pH.
Each of these measurement scales involves
exponential growth.
e.g. An earthquake of magnitude 8 is 32 times more powerful
than an earthquake of magnitude 7.
e.g. A liquid with pH 5 is ten times more acidic than one with
pH 6.
Earthquake Magnitude Scale
The magnitude scale for earthquakes is defined so that
each magnitude represents about 32 times as much
energy as the prior magnitude.
Given the magnitude M we compute the released
energy E using the following formula:
E = (2.5 x 104) x 101.5M
Energy is measured in joules.
Magnitudes have no units.
NOTE: exponential growth
ex1/548 Calculate precisely how much more energy is released for
each 1 magnitude on the earthquake scale (about 32 times more).
Magnitude 1: E = (2.5 x 104) x 101.5(1)
Magnitude 2: E = (2.5 x 104) x 101.5(2)
Magnitude 3: E = (2.5 x 104) x 101.5(3)
For each 1 magnitude, 101.5 = 31.623 times more energy is released.
ex2/548 The 1989 San Francisco earthquake, in which 90 people were
killed, had magnitude 7.1. Compare the energy of this earthquake to
that of the 2003 earthquake that destroyed the ancient city of Bam,
Iran, which had magnitude 6.3 and killed an estimated 50,000 people.
SF in 1989 with M=7.1: E = (2.5 x 104) x 101.5(7.1)
= 1.11671E15 joules
Iran in 2003 with M=6.3: E = (2.5 x 104) x 101.5(6.3)
= 7.04596E13 joules
Since 1.11671E15/7.04596E13 = 15.8489464, we say that:
the SF quake was about 16 times more powerful than the Iran quake.
Earthquake Magnitude Scale
Given the magnitude M we compute the released
energy E (joules) using the following formula:
E = (2.5 x 104) x 101.5M
Given the released energy E, how do we compute the
the magnitude M?
Use common logarithms
Common Logarithms (page 531)
log10(x) is the power to which 10 must be raised to obtain x.
log10(x) recognizes x as a power of 10
log10(x) = y if and only if 10y = x
log10(1000) = 3
since 103 = 1000.
log10(10,000,000) = 7 since 107 = 10,000,000.
log10(1) = 0 since 100 = 1.
log10(0.1) = -1 since 10-1 = 0.1.
log10(30) = 1.4777 since 101.4777 = 30. [calculator]
Common Logarithms (page 531)
Practice with Logarithms (page 533)
23/533 100.928 is between 10 and 100.
25/533 10-5.2 is between 100,000 and 1,000,000.
27/533
log10 ( )
29/533
log10(1,600,000) is between 6 and 7.
31/533
log10(0.25) is between 0 and 1.
is between 0 and 1.
Properties of Logarithms
(page 531)
log10(x) is the power to which 10 must be raised to obtain x.
log10(x) recognizes x as a power of 10
log10(x) = y if and only if 10y = x
log10(10x) = x
log10 ( x )
10
x
log10(xy) = log10(x) + log10(y)
log10(ab) = b x log10(a)
Practice (page 533)
Earthquake Magnitude Scale
Given the magnitude M we compute the released
energy E (joules) using the following formula:
E = (2.5 x 104) x 101.5M
Given the released energy E, how do we compute the
the magnitude M?
log10E = log10[(2.5 x 104) x 101.5M]
log10E = log10(2.5 x 104) + log10(101.5M)
log10E = 4.4 + 1.5M
Earthquake Magnitude Scale
Given the magnitude M we compute the released
energy E (joules) using the following formula:
E = (2.5 x 104) x 101.5M
exponential
Given the released energy E (joules), we compute the
the magnitude M using the following formula:
log10E = 4.4 + 1.5M
logarithmic
More Practice 24*/554: E = 8 x 108 joules
Typical Sounds in Decibels
Decibels
Times Louder
than Softest
Audible Sound
Example
140
1014
jet at 30 meters
120
1012
strong risk of damage to ear
100
1010
siren at 30 meters
90
109
threshold of pain for ear
80
108
busy street traffic
60
106
ordinary conversation
40
104
background noise
20
102
whisper
10
10
rustle of leaves
0
1
threshold of human hearing
-10
0.1
inaudible sound
decibels increase by 10s and intensity is multiplied by 10s.
Decibel Scale for Sound
The loudness of a sound in decibels is defined by the
following equivalent formulas:
intensity of the sound
loudness in dB = 10 log10 ( intensity
of softest audible sound )
loudness in dB
intensity of sound
10 10
intensity of softest audible sound
KEY: How does sound compare to softest audible sound?
More Practice
25/554 How many times as loud as the softest audible
sound is the sound of ordinary conversation? Verify the
decibel calculation on page 549.
27/554 What is the loudness, in decibels, of a
sound 20 million times as loud as the softest
audible sound?
29*/554 How much louder (more intense) is a 25dB sound than a 10-dB sound?
pH Scale for Acidity
The pH is used by chemists to classify substances
as neutral, acidic, or basic/alkaline.
Pure Water is neutral and has a pH of 7.
Acids have a pH lower than 7.
Bases have a pH higher than 7
Solution
pH
Solution
pH
Pure water
7
Drinking water
6.5-8.2
Stomach acid
2-3
Baking soda
8.4
Vinegar
3
Household
ammonia
10
Lemon Juice
2
Drain opener
10-12
The pH Scale
The pH Scale is defined by the following equivalent
formulas:
pH = -log10[H+]
or
[H+] = 10-pH
where [H+] is the hydrogen ion concentration in moles per liter.
Practice
37/555 What is the hydrogen ion concentration of a solution
with pH 7.5?
39/555 What is the pH of a solution with a hydrogen ion concentration
of 0.01 mole per liter? Is this solution an acid or base?
Homework
Pages 554 - 555
# 22,24, 28, 38,40