What is the smallest positive integer a greater than 1, such that the following equation has a positive integer solution in x?
\(\log_a ( x \log_a ( x \log_a ( x \cdots ) ) ) = x\)
+9519 Noticing the repeating pattern is equal to x.
\(\log_a(x\color{red}\log_a(x\log_a(x\cdots))\color{black}) = x\\ \log_a(x\color{red}x\color{black}) = x\\ \log_a(x^2) = x\\ 2\log_a(x)=x\)
By a simple trial-and-error approach,
(a, x) = (2, 2) is a solution.
Therefore the required positive integer a is 2.
