XlogX=4 or logX^X=4 and X^X=b^4, where b is the base of the log. So X=4 if b=4.
When b=10, X^X=10000.
Substitute X=5 and we get 5^5=3125; put X=6 and we get 6^6=46656. The solution is between 5 and 6.
The log of a number is much smaller than the number itself. Write X=4/logX.
Substitute X=5, so log5=0.6990 and the next estimate of X=4/0.6990=5.722.
Log 5.722=0.7575 and the next estimate is X=4/0.7575=5.281.
If we continue with these iterations we arrive at X=5.4385826959. This takes only a minute or two on a calculator.