Exponential Growth Equations

The number of bacteria on Agar plate A is given by the equation n=400(2^d), where d is equal to the number of cellular divisons each cell has undergone. If the cells divides every 2 minutes, how many seconds will it take for the number of bacteria in the agar plate to increase tenfold?

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The first and trickiest thing to realise here is that you can work out the number of bacteria before any divisions have taken place by inputting d=0

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n=400(2^0)
n=400

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Once you have made this realisation the rest of the question is quite simple. You first multiply the number of initial bacteria by ten to workout the value you need to input into the equation modelling bacteria, then proceed to substitute that value into the equation and isolate the exponential term so you can use your log laws

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400*10 = 400(2^d)

4000=400(2^d)

4000/400=2^d

10=2^d

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To solve the simplified equation above for d, we take the log (base 2) of both sides. This gives us the number of divisions. To turn the number of divisions into the amount of minutes that number of divisions takes we multipy the number of divisions by the number of minutes per divison (this is given in the question as two). To convert the number of minutes into the number of seconds just multiply by 60 and we have the answer!

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log2(10)=log2(2^d)
3.3219=d
total time in minutes=3.3219*2
                              =6.6439

Total time in seconds=6.6439*60
                                =398.63