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 Fortieth logs (Posted on 2011-05-31)
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.

 See The Solution Submitted by Jer Rating: 4.5000 (2 votes)

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 re(2): Closer number, closing remarks | Comment 13 of 15 |
(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
 Posted by Ady TZIDON on 2011-06-02 01:59:01

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