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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: Closer number | Comment 12 of 15 |
(In reply to Closer number by Math Man)

The 33/40 approximation was arrived at by saying that 21 is approximately 20.  These two numbers are 5% different.

The 34/40 approximation was arrived at by saying that 49 is approximately 50.  These two numbers are only 2% different.

So, it is not surprising that the 34/40 approximation would be closer (in this case, about 4 times as close).  It was not guaranteed that it would be closer, because there were other approximations involved, but it was very likely.

 Posted by Steve Herman on 2011-06-02 00:42:38

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