# Section 8D Logarithm Scales: Earthquakes, Sounds, and Acids

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Section 8D Logarithm Scales: Earthquakes, Sounds, and Acids
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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, = times more energy is released.

ex2/548 The 1989 San Francisco earthquake, in which 90 people were killed, had magnitude 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) = E15 joules Iran in 2003 with M=6.3: E = (2.5 x 104) x 101.5(6.3) = E13 joules Since E15/ E13 = , 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) = since = 30. [calculator]

Common Logarithms (page 531)

Practice with Logarithms (page 533)
23/ is between 10 and 100. 25/ is between 100,000 and 1,000,000. 27/ is between 0 and 1. 29/533 log10(1,600,000) is between 6 and 7. 31/533 log10(0.25) 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(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 = M

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 = M logarithmic More Practice 24*/554: E = 8 x 108 joules

Typical Sounds in 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 rustle of leaves 1 threshold of human hearing -10 0.1 inaudible sound decibels increase by 10s and intensity is multiplied by 10s.

KEY: How does sound compare to softest audible sound?
Decibel Scale for Sound The loudness of a sound in decibels is defined by the following equivalent formulas: 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 25-dB 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 Pure water 7 Drinking water Stomach acid 2-3 Baking soda 8.4 Vinegar 3 Household ammonia 10 Lemon Juice 2 Drain opener 10-12

The pH Scale pH = -log10[H+] or [H+] = 10-pH
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 # 22,24, 28, 38,40