If $\log_{25} [5 \log_3 (1+\log_3(1+2\log_2x))] = \frac12$ then $x$ is:

We start by simplifying the outer logarithm:

If $\log_{25}(y) = \frac{1}{2}$, then $y = 25^{\frac12} = 5$.

$5 \log_3 (1+\log_3(1+2\log_2x)) = 5$

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Dividing both sides by 5:

$\log_3 (1+\log_3(1+2\log_2x)) = 1$

Since $\log_3(3) = 1$, we have:

$1+\log_3(1+2\log_2x) = 3$

$\therefore \log_3(1+2\log_2x) = 2$

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Since $\log_3(9) = 2$, we have:

$1+2\log_2x = 9$

Therefore: $2\log_2x = 8$

So: $\log_2x = 4$

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$\log_2x = 4$ means $x = 2^4 = 16$

Therefore, $x = 16$ is our answer.

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Verification: Substituting $x = 16$ back into the original equation confirms our answer.

Hence, option two: $x=16$, is the correct answer