(log6(x))^2+6^(log6(x))^2-log6(6x^2)=x^(log6(x))+log6(x/216)

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We can simplify the given expression step by step as follows:

Use the property of logarithms, log_a(b^c) = c * log_a(b), to simplify the expression:

(log6(x))^2 + 6^(log6(x))^2 - log6(6x^2) = x^(log6(x)) + log6(x/216)
(log6(x))^2 + (log6(6)) (log6(x))^2 - (log6(6) + log6(x^2)) = x^(log6(x)) + log6(x/216)
(log6(x))^2 + (log6(6)) (log6(x))^2 - (log6(6) + 2 * log6(x)) = x^(log6(x)) + log6(x/216)

Simplify the terms involving the logarithms using the properties of logarithms:

(log6(x))^2 + (1) (log6(x))^2 - (1 + 2 log6(x)) = x^(log6(x)) + log6(x/216)
(log6(x))^2 + (log6(x))^2 - 1 - 2 log6(x) = x^(log6(x)) + log6(x/216)
2 (log6(x))^2 - 1 - 2 * log6(x) = x^(log6(x)) + log6(x/216)

Convert the expressions involving logarithms to exponential form:

2 x^2 - 1 - 2 log6(x) = x^x + log6(x/216)

This is the simplified form of the given expression.