The mantissa of the common log of 31 is near .5 as that pair of leading digits is very near the middle of the C or D scale of a slide rule. The characteristic is 1 as it's a 2digit number. So the log of 31^11 is about 11*1.5 = 16.5.
The mantissa of the log of 2 is about .30. Interpolating from the log of 1, we'd get about (7/10)*.3 = .21, and actually it should be somewhat larger due to logs slope decreasing with higher numbers, but we'll use .21 as conservative. The characteristic for 17 is again 1. So log(17^14) is conservatively 14*1.21, which is larger than 14*1.2 = 16.8.
So log(17^14) > log(31^11) and therefore 17^14 > 31^11.

Posted by Charlie
on 20130709 12:23:15 