To solve the given equation log9⁡(3log2⁡(1 log3⁡(1 2log2⁡x))) = 1/2, we need to simplify the expression step by step.

Step 1: Simplify the innermost expression
log3⁡(1 2log2⁡x)
Using the logarithmic identity loga⁡(mn) = loga⁡m + loga⁡n, we can rewrite the expression as:
log3⁡(1) + log3⁡(2log2⁡x)
Simplifying further, we get:
0 + log3⁡(2log2⁡x)
= log3⁡(2log2⁡x)

Step 2: Simplify the next innermost expression
3log2⁡(1 log3⁡(1 2log2⁡x))
Using the logarithmic identity loga⁡(mn) = loga⁡m + loga⁡n, we can rewrite the expression as:
3(log2⁡(1) + log2⁡(log3⁡(1 2log2⁡x)))
Simplifying further, we get:
3(0 + log2⁡(log3⁡(1 2log2⁡x)))
= 3log2⁡(log3⁡(1 2log2⁡x))

Step 3: Simplify the remaining expression
log9⁡(3log2⁡(1 log3⁡(1 2log2⁡x)))
Using the logarithmic identity loga⁡(mn) = loga⁡m + loga⁡n, we can rewrite the expression as:
log9⁡(3) + log9⁡(log2⁡(log3⁡(1 2log2⁡x)))
Simplifying further, we get:
log9⁡(3) + log9⁡(log2⁡(log3⁡(1 2log2⁡x)))

Step 4: Apply the change of base formula
To simplify further, we can convert the base of logarithms to a common base, such as 10.
log9⁡(3) + log9⁡(log2⁡(log3⁡(1 2log2⁡x)))
= log(3)/log(9) + log(log2⁡(log3⁡(1 2log2⁡x)))/log(9)

Step 5: Simplify the expression involving logarithms
Using the logarithmic identity loga⁡(b/c) = loga⁡b - loga⁡c, we can rewrite the expression as:
log(3)/log(9) + (log(log2⁡(1 2log2⁡x)) - log(log3⁡(1 2log2⁡x)))/log(9)
= log(3)/log(9) + (log(log2⁡(1) + log2⁡(2log2⁡x)) - log(log3⁡(1) + log3⁡(2