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

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.