It turns out that the common logarithms of each of the numbers from 2 through 9 can be very well approximated by rational numbers of the form n/40.

Derive each of the numerators with no calculation aids beyond pencil and paper.

(In reply to A start (spoiler) by Steve Herman)

log 5 = log 10 - log 2 ~= 1 - 12/40 = 28/40

and perhaps for 7 we just need to interpolate between 31/40 for 6 and 36/40 for 8, with 34/40 giving the extra half point to the lower side (a 3-2 split instead of 2-3) as the intervals between logs get lower as the numbers get higher.

Edited on May 31, 2011, 2:50 pm

Posted by Charlie on 2011-05-31 14:43:51