<s>[INST] A bacterial culture produces new bacteria at a rate that is proportional to the number of bacteria present. If the initial amount of the bacteria is $P$ and after some time the amount becomes $Q$, then the time taken for this change, say $T$, is given by $$T = k\log \frac{Q}{P}$$ where $k$ is some constant. This problem is basically asking us to determine a certain time interval during which the number of bacteria may become double of the initial number.[/INST]