Question 387793
first, let us make a table to see how the data behaves:

1/2  hour periods                                   0          1          2          3          4 .......

number of bacteria                              90        180      360    720     1440

As stated in the problem, the above shows that the initial number of bacteria doubles each 1/2 hour. It can also be seen that we can calculate the number of bacteria reached at any period by multiplying the original amount of 90 by 2^period number. For example, the 720 reached after 3 periods is determined by multiplying the original amount of 90 by 2^3, after 4 periods by 2^4, after 2 periods by 2^2, etc.   

If we call the original amount, P, and the amount reached after a certain number of 1/2 hour periods, A, and period number, n, we can come up with the following relationship:

A =  P (2^n)

For this problem:

368640 = 90(2^n)
2^n=368640/90=4096
use logarithms to solve
n (log 2)=log 4096
n=log 4096/(log 2) = 12

ans: after 12 (1/2 hour) periods, or 6 hours, the bacteria count would have reached 368640