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 re: Closer number
by Steve Herman)
No doubt, you are right.
That is my reason for taking 2401 instead of 2400.
However , we accrue errors by starting with log 2 (1024 instead of 1000 ),then err in the other direction (80 instead 81), err again(20 instead of 21 or 100 instead of 98 ) in case of log 7, loosing track of the accuracy of our results. One can get a better approximation of log2 e.g. by succesive extractions of square root of 10, approaching 1+h, where h can be made as small as desired, but this is hardly a "pen & paper" operation.
If anyone wants to improve the n/40 results, there are better starting points, like searching for expressions like p^a*q^b *r^c being close to 10^d (p ,q distinct prime digits, a,b,c,d integers, not necessarily positive).
Taking care of the accuracy evaluation at each stage of the approximation process, you can reach the pre-defined resolution.
I got some very good results.
Edited on June 2, 2011, 2:05 am