In a book of math puzzles Peter Winkler discusses the sequence x, x^x, x^x^x, x^x^x^x and writes about conditions for its divergence. Clearly for x=1 the sequence is 1,and for x=2 it diverges to infinity. Then he shows that the maximum value of x for the sequence to converge is x=e^(1/e) or x= 1.44467... At this value the infinite tower of "x" exponents is equal to e (2.7182818..). For any x larger than e^(1/e), the sequence diverges to infinity. Mr Winkler later goes on to discuss the equation x^x^x^x^x...=2 (an infinite tower of "x" exponents=2) By using the trick that the exponent of the bottom"x" is the same as the whole expression,the equation becomes x^2=2, and x=sqrt(2)=1.414... is the solution. (This is close to the maximum value for convegence( shown above) 1.44467... My question(finally): Suppose you have an equation x^x^x^x^x^x^x...=10 (an infinite tower of "x"=10) Why can't you use the same trick as we did for x^x^x^x^x^x=2 case. In this case we would get x^10=10. And the solution is x= the tenth root of 10(x=1.2589..) Now the tenth root of 10 is clearly the wrong answer because: 1) It is too small . It is smaller than sqrt(2) whose tower conveges to the number 2 2) The maximum value this tower of "x" converges to is 2.71828... at x=e^(1/e) Why doesn't this trick work for x^x^x^x^x^x...=10 ?