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.

Well, now that we know log 5,

7^2 = 49 ~ 50

so 2log 7 ~ log 50 = 1 + log 5 ~ 1+ 28/40 = 68/40

==> log 7 ~ 34/40