If value halves every $5$ years, when will the dollar be worth $1/1,000,000$ its current value?

Question

This was a GRE multiple choice question.

At a $15$ percent annual inflation rate, the dollar would decrease by approximately one-half every $5$ years. At this inflation rate, in approximately how many years would the dollar be worth $\dfrac{1}{1,000,000}$ of its present value?

(A) $25$

(B) $50$

(C) $75$

(D) $100$

(E) $125$

The correct answer is (D), and I solved this by running through powers of $2$ and noting $2^{20}\approx 1,000,000$, so it would take around $20\cdot 5=100$ years. Is there a less haphazard way of setting this problem up to solve it?

Answer 1

The explicit way is to say you want $(\frac 12)^n \lt \frac 1{1000000}, 2^n \gt 1000000$ and take the base 2 log of both sides, getting $n \gt \log_2 1000000=6(\log_2 10) \approx 19.9316$ so $n=20$ is the lowest integer.

A less formal way is to notice that $2^{10}=1024 \gt 1000$ so $2^{20} \gt 10^6$

Answer 2

$2^{10} = 1024$ and $1024^2 \approx 1,000,000$ and so, you get $2^{20}$.