oVlad wrote: Let $a\neq 1$ be a positive real number. Find all real solutions to the equation $a^x=x^x+\log_a(\log_a(x)).$ Mihai Opincariu I suppose that $x>0$ (in order $x^x$ be defined without studying specific cases of integers or some rational numbers). If $x>a$ : $x^x>a^x$ and $\log_a\log_ax>\log_a \log_a a=0$ and so $RHS>LHS$ If $x<a$ : $x^x<a^x$ and $\log_a\log_ax<\log_a \log_a a=0$ and so $RHS<LHS$ And so only positive possibility is $\boxed{x=a}$, which indeed fits.

pco wrote: I suppose that $x>0$ (in order $x^x$ be defined without studying specific cases of integers or some rational numbers). We indeed have $x>0$ in order for $\log_ax$ to be well-defined.