When the colony outgrows its life support

EasyBrainteasers Calculus~5m

A space station's population quadruples every $54$ years, while its life-support capacity only doubles every $54$ years. Initially the population is $2$ and the capacity is $16384$ . After how many years does the population first exceed the capacity?

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In $k$ periods: population $=2\cdot4^k=2^{2k+1}$ , capacity $=2^{14}\cdot 2^k$ .
Solve $2k+1>14+k\Rightarrow k>13$ ; the answer is $14\times 54$ .

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When the colony outgrows its life support

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