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

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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.

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The magnitude scale for earthquakes is defined so that each magnitude represents about 32 times as much energy as the prior magnitude. Earthquake Magnitude Scale Given the magnitude M we compute the released energy E using the following formula: E = (2.5 x 10 4 ) x M Energy is measured in joules. Magnitudes have no units. NOTE: exponential growth

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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 10 4 ) x (1) Magnitude 2: E = (2.5 x 10 4 ) x (2) Magnitude 3: E = (2.5 x 10 4 ) x (3) For each 1 magnitude, = times more energy is released.

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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 10 4 ) x (7.1) = E15 joules Iran in 2003 with M=6.3: E = (2.5 x 10 4 ) x (6.3) = E13 joules Since E15/ E13 = , we say that: the SF quake was about 16 times more powerful than the Iran quake.

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Earthquake Magnitude Scale Given the magnitude M we compute the released energy E (joules) using the following formula: E = (2.5 x 10 4 ) x M Given the released energy E, how do we compute the the magnitude M? Use common logarithms

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Common Logarithms (page 531) log 10 (x) is the power to which 10 must be raised to obtain x. log 10 (x) recognizes x as a power of 10 log 10 (x) = y if and only if 10 y = x log 10 (1000) = 3 since 10 3 = log 10 (10,000,000) = 7 since 10 7 = 10,000,000. log 10 (1) = 0 since 10 0 = 1. log 10 (0.1) = -1 since = 0.1. log 10 (30) = since = 30. [calculator]

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Common Logarithms (page 531)

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

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Properties of Logarithms (page 531) log 10 (10 x ) = x log 10 (xy) = log 10 (x) + log 10 (y) log 10 (a b ) = b x log 10 (a) Practice (page 533) log 10 (x) is the power to which 10 must be raised to obtain x. log 10 (x) recognizes x as a power of 10 log 10 (x) = y if and only if 10 y = x

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Earthquake Magnitude Scale Given the magnitude M we compute the released energy E (joules) using the following formula: E = (2.5 x 10 4 ) x M Given the released energy E, how do we compute the the magnitude M? log 10 E = log 10 [(2.5 x 10 4 ) x M ] log 10 E = log 10 (2.5 x 10 4 ) + log 10 (10 1.5M ) log 10 E = M

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Earthquake Magnitude Scale Given the magnitude M we compute the released energy E (joules) using the following formula: E = (2.5 x 10 4 ) x M Given the released energy E (joules), we compute the the magnitude M using the following formula: log 10 E = M More Practice 24*/554: E = 8 x 10 8 joules exponential logarithmic

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Typical Sounds in Decibels DecibelsTimes Louder than Softest Audible Sound Example jet at 30 meters strong risk of damage to ear siren at 30 meters threshold of pain for ear busy street traffic ordinary conversation background noise whisper 10 rustle of leaves 01threshold of human hearing inaudible sound decibels increase by 10s and intensity is multiplied by 10s.

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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?

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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 /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?

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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 SolutionpHSolutionpH Pure water7Drinking water Stomach acid2-3Baking soda8.4 Vinegar3Household ammonia 10 Lemon Juice2Drain opener10-12

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The pH Scale The pH Scale is defined by the following equivalent formulas: pH = -log 10 [H + ] or [H + ] = 10 -pH where [H + ] is the hydrogen ion concentration in moles per liter.

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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?

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Homework Pages # 22,24, 28, 38,40

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