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# hard equation

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What is the smallest positive integer a greater than 1, such that the following equation has a positive integer solution in x?

\(\log_a ( x \log_a ( x \log_a ( x \cdots ) ) ) = x\)

Jun 19, 2020

#1
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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.

Jun 19, 2020