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.

7 = 21/3

So, log 7 = log 21 - log 3

~ log 20 - log 3 = 1 + log 2 - log 3

= 1 + 12/40 - 19/40

= 33/40

Great problem! :-)