Why should we choose to use 5^3+ instead of 5^4-? I would probably use 5^4- since this is closer to 512. Now if 5^4- is chosen, then it would be 11- < n, which would lead to answer choice D. The method presented in the video seems to be flawed.
The key here is in the logic behind why we chose 5^3+ instead of 5^4-, though ultimately it doesn't actually matter. Alright, so when we're looking at 512 and wondering what power of 5 would be the closest, we quickly see our possibilities:
5^3 = 125
5^4 = 625
So it must be somewhere between 3 and 4. If that's the case 512 must be greater than 5^3. That's extremely important to recognize. So then we go on to add:
5^(some # > 3) + 5^7 = 5^(some # > 3 + 7) = 5^(some # > 10)
The thing is, even if we had chosen 5^4- as our estimation, it wouldn't matter, because we logically know that even if the number is less than 4, it's still greater than 3. So then we'd have:
5^(some # < 4) + 5^7 = 5^(some # < 4 + 7) = 5^(some # <11)
So our result is that we're looking for a number between 10 and 11, therefore it is greater than 10. Since the question asks for the least possible value of integer n that renders 5^n greater than 40,000,000, and the actual value is somewhere between 10 and 11, it has to be more than 10, and is therefore 11. To test that, here's the actual solution:
5^10 = 9,765,625
5^11 = 48,828,125
This proves our previous work.