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 re(2): Who is right for 7?....DETAILS
1,By hand- aided by excel to get the powers evaluated;
2, You can get log7, provided you know log3.
3. There is a multitude of combinations, too long to list- I have just summed up the guidelines.