Nested logarithmic equation with a parameter

$\begin{aligned} \log_a(x\log_a(x\log_a(x \cdots))) & = x \\ \log_a(x^2) & = x \\ a^x & = x^2 \end{aligned}$

There are two cases, when $x = \begin{cases} 2 & \implies a = 2 \\ 4 & \implies a = 2 \end{cases}$ . Therefore $a = \boxed 2$ .

$\log_{a}(x \cdot \boxed{\log_{a}(x \cdot \log_{a}(x \cdot \cdot \text{ } \cdot))})=x$

We can take advantage of the fact that a copy of the outer expression exists inside the expression. The boxed value must be $x$ .

$\log_{a}(x \cdot x)=x$ $\log_{a}(x^{2})=x$ $2\log_{a}(x)=x$ $\log_{a}(x)=\dfrac{x}{2}$

We need to find the smallest natural value for $a$ such that $a>1$ . When we try $a=2$ , we can use a graphing calculator or a software like Desmos to show that the two graphs cross, and solutions exists. Thus, $\boxed{a=2}$ .