SOLUTION: I need your help on the following word problem: Given a starting population of 100 bacteria, the formula b=100(2^t) can be used to find the number of bacteria, "b", after "t" peri

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Question 190620: I need your help on the following word problem:
Given a starting population of 100 bacteria, the formula b=100(2^t) can be used to find the number of bacteria, "b", after "t" periods of time. If each period is 15 minutes long, how many minutes will it take for the population of bacteria to reach 51,200?
Answer by jim_thompson5910(28536) About Me  (Show Source):
You can put this solution on YOUR website!
b=100%282%5Et%29 Start with the given equation.


51200=100%282%5Et%29 Plug in b=51200 (the given bacteria population)


512=2%5Et Divide both sides by 100.


log%2810%2C%28512%29%29=log%2810%2C%282%5Et%29%29 Take the log base 10 of both sides


log%2810%2C%28512%29%29=t%2Alog%2810%2C%282%29%29 Rewrite right side using the identity log%28b%2C%28x%5Ey%29%29=y%2Alog%28b%2C%28x%29%29


log%2810%2C%28512%29%29%2Flog%2810%2C%282%29%29=t Divide both sides by log%2810%2C%282%29%29


log%282%2C%28512%29%29=t Use the change of base formula to rewrite the left side


Note: Remember the change of base formula is log%28b%2C%28x%29%29=log%2810%2C%28x%29%29%2Flog%2810%2C%28b%29%29


t=log%282%2C%28512%29%29 Rearrange the equation


t=9 Evaluate the log base 2 of 512 to get 9


Note:


since 2%5E9=512, this means that log%282%2C%28512%29%29=9


So the answer is t=9 which means that it will take 9 minutes for the population to reach 51,200