I couldnt find the exact solution, not even with the help of mathematica. That mathematica gave me one of the most complicated solution. Anyway I tried to slove it in analytical way for the general case, x^a=a^x ........(lets consider a is an integer and |a|>0) log(a)(x^a)=log(a)(a^x)..(this is not natural logarihtm it is the logarithm with base a) alog(a)(x)=xlog(a)(a) alog(a)(x)=x.........(cause log(a)(a)=1) a=x/log(a)(x).......................(1) From equation 1 we can conclude following things. 1.x is an integer. 2.value of x follows the following progression(a,a^a,a^(a^a) and so on) So it can be concluded from here that this equation can have two real roots only if a^(a^(a^.........)) upto nth term/a^(a^(a^..........)) upto n-1th term=a otherwise there is only one real root which is equal to a. Hehe hope someone can come with better solution.