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Orders of Magnitude Order of Magnitude – the common logarithm of a positive quantity. Examples Mercury is about 5.79 x 10 meters from the Sun 10 Pluto is about 5.9 x 10 meters from the Sun 12 Find the common logarithm of both numbers, and then calculate the difference between these values… Pluto’s distance from the Sun is 2 orders of magnitude greater than Mercury’s magnitude greater than Mercury’s

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Orders of Magnitude Order of Magnitude – the common logarithm of a positive quantity. Determine how many orders of magnitude the quantities differ. 1. A kilometer and a meter 3 3 3 3 2. A $1 dollar bill and a penny 2 2 2 2 3. A horse weighing 400 kg and a mouse weighing 40 g 4 4 4 4 4. NYC with 7 million people and Earmuff Junction with a population of 7 6 6 6 6 Examples

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The level of sound intensity in decibels (dB) is Application: Decibels where (beta) is the number of decibels, I is the sound intensity in W/m, and I = 10 W/m is the threshold of human hearing (the quietest audible sound intensity). 2 0 –122

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The level of sound intensity in decibels (dB) is Application: Decibels The sound of a subway train is 100 dB, and the sound of a soft whisper is 10 dB. By how many orders of magnitude do these quantities differ? We seek the logarithm of the ratio

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Application: Decibels The two sound intensities differ by 9 orders of magnitude!!!

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Application: Richter Scale The Richter Scale magnitude R of an earthquake is where a is the amplitude in micrometers of the vertical ground motion at the receiving station, T is the period of the associated seismic wave in seconds, and B accounts for the weakening of the seismic wave with increasing distance from the epicenter of the earthquake.

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Application: Richter Scale How many more times severe was the 2001 earthquake it Gujarat, India (R = 7.9) than the 1999 earthquake in Athens, Greece (R = 5.9)? 1 2 We seek the ratio of severities

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Application: Richter Scale A Richter scale difference of 2 corresponds to an amplitude ratio of 100 The Gujarat quake was 100 times as severe as the Athens quake!!!

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Application: Chemical Acidity The measure of acidity is pH, the opposite of the common log of the hydrogen-ion concentration of a solution: Note: More acidic solutions have higher hydrogen-ion concentrations, and therefore have lower pH values…

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Application: Chemical Acidity Some especially sour vinegar has pH of 2.4, and a box of baking soda has a pH of 8.4. (a) What are their hydrogen-ion concentrations? Vinegar: moles per liter Baking Soda: moles per liter

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Application: Chemical Acidity Some especially sour vinegar has pH of 2.4, and a box of baking soda has a pH of 8.4. (b) How many times greater is the hydrogen-ion concentration of the vinegar than that of the baking soda? of vinegar of baking soda times greater

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Application: Chemical Acidity Some especially sour vinegar has pH of 2.4, and a box of baking soda has a pH of 8.4. (c) By how many orders of magnitude do the concentrations differ? The hydrogen-ion concentration of the vinegar is 6 orders of magnitude greater than that of the baking soda… This is exactly the difference in their pH values!!!

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Application: Newton’s Law of Cooling An object that has been heated will cool to the temperature of the medium in which it is placed (such as the surrounding air or water). The temperature T of the object at time t can be modeled by Newton’s Law of Cooling: Temp. of the surrounding medium Initial Temp. of the object A constant

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Application: Newton’s Law of Cooling A hard-boiled egg at temperature 96 C is placed in 16 C water to cool. Four minutes later the temperature of the egg is 45 C. Use Newton’s Law of Cooling to determine when the egg will be 20 C. Identify terms: Plug into equation: Use the point (4, 45) to solve for k:

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Application: Newton’s Law of Cooling A hard-boiled egg at temperature 96 C is placed in 16 C water to cool. Four minutes later the temperature of the egg is 45 C. Use Newton’s Law of Cooling to determine when the egg will be 20 C. Save this value in your calculator!!!

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Application: Newton’s Law of Cooling A hard-boiled egg at temperature 96 C is placed in 16 C water to cool. Four minutes later the temperature of the egg is 45 C. Use Newton’s Law of Cooling to determine when the egg will be 20 C. Finally, solve this equation: The temp. of the egg will be 20 C after about 11.8 minutes

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Properties of Logarithms By: Jennifer Garcia & Roslynn Martinez.

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