What is solution of this expression

Question

I am new to logarithms, and I need to find out the solution set of this expression.

$$x^{\log_a x} = (a^\pi)^{(\log_a x)^3} \\ a \in \mathbb{N} , a>0 ,a \neq 1$$

Answer 1

Let $\log_a x=t$, then $a^t=x$. Use this in the equation given to write \begin{align*} x^{\log_a x} & = (a^\pi)^{(\log_a x)^3}\\ x^{t}&=a^{\pi t^3}\\ a^{t^2} & = a^{\pi t^3}\\ t^2&=\pi t^3. \end{align*} Thus either $t=\log_a x=0$ or $t=\log_a x=\frac{1}{\pi}$. This means $x=1$ or $x=a^{\frac{1}{\pi}}$