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SOML MEET 4 EVENT 2 Log and Exponential Functions 1. [2 Points] Suppose NAME: _____________ TEAM: _____________ SCHOOL: _____________ log b Y = log c Z = x Find log 2 and b = 2. c Y . Z ANS: _____________ 2. [3 Points] I ,where I is the intensity S of the earthquake (measured by the amplitude of a seismography reading taken 100km from the epicenter of the earthquake) and S is the intensity of a “standard” earthquake (whose amplitude is 1 micron = 10 −4 cm ). The 1906 earthquake in San Francisco had an estimated magnitude of 8.3 on the Richter scale. In the same year a powerful earthquake occurred on the Columbian-Ecuador border and was four times as intense. What was the magnitude of the Columbia-Ecuador earthquake on the Richter scale, to the nearest tenth? The magnitude M of an earthquake is: M = log ANS: _____________ 3. [5 Points] An infection strain of bacteria grows by doubling every hour. When a certain critical number of bacteria are present in the bloodstream, a person becomes ill. If a single bacterium infects a person, the critical level is reached in 24 hours. How long will it take for the critical level to be reached if the same person is infected with 10 bacteria? Give your answer to the nearest hundredth of an hour. ANS: ___________hrs SOML MEET 4 EVENT 2 Log and Exponential Functions 1. [2 Points] Solution: Suppose NAME: ___KEY______ TEAM: _____________ SCHOOL: _____________ log b Y = log c Z = x and Y b = 2 . Find log 2 . c Z log 2 Y = log 2 2 x = x. Z log b Y = x ⇒ b x = Y log c Z = x ⇒ c x = Z x Y bx ⎛ b ⎞ = x = ⎜ ⎟ = 2x ; Z c ⎝c⎠ ANS: _____x______ 2. [3 Points] The magnitude M of an earthquake is: M = log I , where I is the intensity of the S earthquake (measured by the amplitude of a seismography reading taken 100km from the epicenter of the earthquake) and S is the intensity of a “standard” earthquake (whose −4 amplitude is 1 micron = 10 cm ). The 1906 earthquake in San Francisco had an estimated magnitude of 8.3 on the Richter scale. In the same year a powerful earthquake occurred on the Columbian-Ecuador border and was four times as intense. What was the magnitude of the Columbia-Ecuador earthquake on the Richter scale, to the nearest tenth? Solution: If I is the intensity of the San Francisco earthquake, then from the definition of magnitude we have: I M = log S = 8.3 The intensity of the Columbian-Ecuador earthquake was 4I, so its magnitude was: M = log 4. [5 Points] 4I = log 4 + 8.3 = 8.9 S ANS: _____8.9____ An infection strain of bacteria grows by doubling every hour. When a certain critical number of bacteria are present in the bloodstream, a person becomes ill. If a single bacterium infects a person, the critical level is reached in 24 hours. How long will it take for the critical level to be reached if the same person is infected with 10 bacteria? Give your answer to the nearest hundredth of an hour. Solution: B ( t ) = B0 2t . When B0 = 1 , B ( 24 ) = 224 = 16, 777, 216 . When B0 = 10 , we solve 10 ( 2t ) = 16, 777, 216 . 2t = 1, 677, 721.6 → t ln 2 ≈ ln (1, 677, 721.6 ) → t ≈ 20.68 hrs. ANS: _20.68 hours_