I've been pondering this quirky fact about powers of 2 for a while now, and I can't seem to formulate it properly in my head to find a proper answer without just using a calculator and trying out numbers till I find the answer.
When talking about memory we usually use Megs, Gigs, etc. And usually these are technically powers of 2.
Kilo = pow(2,10) Meg = pow(2,20) Gig = pow(2,30)
Ok so it seems that 2 to the power of something divisible by 10 is going to be the approximation of 10 to the power of something divisible by 3 (which seems coincidental even strange to me), and by approximation I mean the first digit is a 1 and it has the correct number of digits.
Using experimentation it appears pow(2,299) is the first time this doesn't hold true, but I can't help but think there must be a way to make a formula that gives me the value without experimentation.
I know for example that log 2 is .30102... which is the reason it works for sufficiently small exponents since we want 3 extra digits each time. But what's next?