### Number of solutions to the equation ${x^{{x^x}}} = x$

(A) 1     (B) 2     (C) 3     (D) None of the options given

We have, ${x^{\left( {{x^x}} \right)}} = x$

Taking log, ${x^x}\ln |x| = \ln |x|$

$\Rightarrow \ln |x|\left( {{x^x} - 1} \right) = 0$

$\ln |x| = 0,x = \pm 1$ Or ${x^x} = 1$

Taking log, $x\ln |x| = 0$

x = 0 Or $\ln |x| = 0,x = \pm 1$

x = 0 is rejected as it leads to $0^0$ situation.

Hence, two solutions or option (B).