With paper and pencil, make a table of 7^n, n being all positive integers. Look for 7^n being close to a power of 10 for a somewhat large exponent. 7^19 fits this quite well = 113988951853731, which is just greater than 10^16. (Whew). While y=log(x) is not asymptotic, its slope is monotonically decreasing, so one can approximate the log(7^19) by log (10^16)=16. Calculate 16/19=0.842185 to six digits (by long division). This is within 0.003 of the exact answer.
Edited on January 23, 2011, 3:01 pm
Edited on January 23, 2011, 3:03 pm

Posted by Kenny M
on 20110123 14:56:37 