Help: Inequalities and Exponents

4000000 = 2^2 * 1000000 = 2^8 * 5^6 = 256 * 5^6.

5^3 = 125
5^4 = 625.

Now 125<256<625.

So we could say 5^n should be 5^4* 5^6 (or) n = 10 D

Nice, Shankar! Another way to think about this, too, is that you know that the GMAT isn't going to require you to actually do that math, so you can use familiar numbers to get an estimate. As Shankar did, you should be able to get to 5^4 = 625 without too much trouble, so that can be your "anchor".

5^8 is going to be 625*625 (5^4 * 5^4), but since we don't want to do 3-digit multiplication let's just estimate with 600*600, which should quite clearly be 360,000. That's clearly too small and not really even close, so we know that making it 625*625 isn't going to be the difference that launches 360,000 up to 4 million.

Even if we multiply 360,000 by 5 (to estimate a little lower than 5^9), we're still looking at less than 2 million (360,000 * 5 = 1,800,000 but you don't even need to go that far). So 5^9 is also too small (again, turning our easy-to-estimate 600*600 into the actual 625*625 isn't enough of a change to more than double 1,800,000 so we don't need to worry because it isn't close).

But if we take that 1,800,000 and multiply by 5 (to simulate 5^10), it's going to be well over 4 million, so we know that 5^10 will get us where we need to be. Through pretty quick estimates we can prove that the least possible integer value of n is 10.

this is a smart n quick way to solve...an otherwise tough question