If $a$ and $x$ are positive numbers and $A=a^2$, express the following in its simplest form in therms of $x$. $a^{\log_a x+\log_{A} x}$

Question

If $a$ and $x$ are positive numbers and $A=a^2$, express the following in its simplest form in terms of $x$. $$a^{\log_{a} x+\log_{A} x}$$ I already know \begin{align} a^{\log_{a} x + \log_{a^2} x} = a^{\log_{a} x} \cdot a^{\log_{a^2} x}, \end{align} but I am not sure how to proceed.

Answer 1

Hint: $$a^{\log_a(y)}=y,$$ and $$\log_{a^2}(y)=\frac{\log_{a}(y)}{\log_{a}(a^2)}=\frac{\log_{a}(y)}{2}.$$

Answer 2

Hints:

(1) By definition, $p^{\log_p q}\equiv q$

(2) $\log_p q = \frac{\log_r q}{\log_r p}$ for any other base $r>0$