Doubling time Formula

[Priest]

Can Anyone help me with this problem, i need a formula to work out how long it takes Bacteria to multiply !

Ok heres the information i start out with:

Initial number of bacteria
End number of bacteria
Time elapsed

assumptions:
Bacteria grow by division into two, therefore each round of growth the number of bacteria double.
There are no deaths and the division is always perfect

So i want to know how long it takes for each division to complete.

eg start with 1 bacteria, end with 4 and total time is 4 hours, then the doubling time would be 2hours - 2 divisions / 4 hours
intuitive, easy ! but whats the formula ?

FYI the actual numbers involved would typically be

start 0.34
end 3.2x10^8
time 144 hours

Help !
TY

[krtxmrtz]

The time it takes for a division cycle is the total time over the number of divisions. The latter are calculated in this fashion:

Let i be the i-eth division step and N(i) the number of bacteria just before that step. Then:

N(2) = 2*N(1) = 2¹*N(1)
N(3) = 2*N(2)=2*2*N(1) = 2²*N(1)
N(4) = 2*N(3) = 2*2*N(2) = 2*2*2*N(1) = 2³*N(1)
...
N(k) = 2^k*N(1)

so after k steps you end up with N(k) bacteria. Then, inverting the formula,

k = log[N(k)/N(1)] / log(2)

and the time per division step is therefore:

t = T / k

where T is the given total time.

Applied to your example:

k = log(3.2*10^8/0.34)/log(2) = 29.810
t = 144 h / 8.974 = 4.831 hours

[Guv]

Are you really starting with 34/100 of a bacterium?

From what you describe, it seems that you should start by dividing the final number of bacterai by the starting number. This should be a power of two, say 2¹⁰ for example. That would indicate ten divisions. Divide into the time.