Is x ^ (log a base x)= a....Why

If yes, please give the proof.

let x^(log a base x) = y

taking log base x on both sides we have,

(log a base x) * (log x base x) = log y base x

but log x base x = 1 for x not equal to 1

thus log a base x = log y base x

hence y = a

hence x^(log a base x) = a.

proved