Consider the system of equations
` log_(3)(log_(2)x)+log_(1//3)(log_(1//2)y) =1 and xy^(2) = 9`.
The value of x in the interval

Options

A. `(200, 300)`
B. ` (400, 500)`
C. `(700, 800)`
D. none of these

Explanation

To solve the given system of equations:

1. **Equations**:
\[ \log_{3}(\log_{2} x) + \log_{1/3}(\log_{1/2} y) = 1 \]
\[ xy^{2} = 9 \]

2. **Rewrite the logarithm**:
We can rewrite \(\log_{1/3}(\log_{1/2} y)\) using the property \(\log_{a}(b) = -\log_{1/a}(b)\):
\[ \log_{3}(\log_{2} x) - \log_{2}(\log_{1/2} y) = 1 \]

3. **Combine the logarithmic terms**:
Using the property \(\log_{a}(b) - \log_{a}(c) = \log_{a}(\frac{b}{c})\):
\[ \log_{3}\left(\frac{\log_{2} x}{\log_{1/2} y}\right) = 1 \]

4. **Convert the logarithmic equation to exponential form**:
From \(\log_{3}\left(\frac{\log_{2} x}{\log_{1/2} y}\right) = 1\), we can write:
\[ \frac{\log_{2} x}{\log_{1/2} y} = 3 \]

5. **Substituting \(\log_{1/2} y\)**:
Since \(\log_{1/2} y = -\log_{2} y\), we substitute:
\[ \frac{\log_{2} x}{-\log_{2} y} = 3 \]
This simplifies to:
\[ -\log_{2} x = 3 \log_{2} y \]
Which can be rewritten as:
\[ \log_{2} x = -3 \log_{2} y \]

6. **Express \(x\) in terms of \(y\)**:
This implies:
\[ x = 2^{-3 \log_{2} y} = y^{-3} \]

7. **Substituting into the second equation**:
Now we substitute \(x = y^{-3}\) into the second equation \(xy^{2} = 9\):
\[ (y^{-3})y^{2} = 9 \]
Simplifying gives:
\[ \frac{y^{2}}{y^{3}} = 9 \implies \frac{1}{y} = 9 \implies y = \frac{1}{9} \]

8. **Finding \(x\)**:
Now substituting \(y = \frac{1}{9}\) back to find \(x\):
\[ x = y^{-3} = \left(\frac{1}{9}\right)^{-3} = 9^3 = 729 \]

9. **Conclusion**:
The value of \(x\) is \(729\).