When a fast-growing colony outruns its life support

A space colony starts with $2$ people and a life-support system that can sustain $16384$ people. Every $54$ years the population multiplies by $4$, while the life-support capacity can only be doubled in the same period. After how many years does the population reach the system's capacity?

$702$ years. After $X$ periods the population is $2\cdot4^X=2^{2X+1}$ and the capacity is $2^{14}\cdot2^X=2^{14+X}$. Equality gives $2X+1=14+X$, so $X=13$ periods, i.e. $13\cdot54=702$ years.