Let k∈Q logk⁡((logk (x)) = log(k^2 )⁡(log(k^2 ) (x)) Solve for x in terms of k

A rational number generalization of ARML 2009 Individual Problem 5 and a tribute to the original deceased(possibly caused by excessive alcohol consumption-suresh bada October 2022)math/physics problem composer and MIT alumnus James Albrecht(Not the Professor of Economics in Georgetown)
At its heart, this problem is a double composition logarithm equation where the logarithm bases are typically relatively small positive integers. The multiplication law of exponents was used twice and the 1-1 property of exponential function gives us permission to equate the exponents leading to the solution x=sqrt(k)