First, we can rewrite $0.0005$ as:

$0.0005 = \frac{5}{10,000}$ = $\frac{1}{2000}$


This question boils down to: for what value of $k$ will $3^k > 2000$?

We know this is possible, so E is incorrect. C and D would work, but we want the least number. We don't need 600+ multiples of 3 to get above 2000, so we can cross those out. We're left with A and B.

$k = 2 => \frac{1}{9}$
$k = 3 => \frac{1}{27}$
$k = 4 => \frac{1}{81}$
$k = 5 => \frac{1}{243}$

At this point, instead of continuing I'd estimate $\frac{1}{243}$ to $\frac{1}{250}$ in order to save time with calculation. Though by all means, if you have time to use a calculator or want to write out the multiplication, that's fine too.

$k = 5 => ~\frac{1}{250}$
$k = 6 => ~\frac{1}{750}$
$k = 7 => ~\frac{1}{2250}$

k = 7 is the least number for $3^k > 2000$, so answer B is correct.