If `xgt2` is a solution of the equation
`|log_sqrt3x-2|+|log_3x-2|=2`, then the value of x is

Answer

To solve the equation \( | \log_{\sqrt{3}} x - 2 | + | \log_{3} x - 2 | = 2 \) given that \( x > 2 \), we will follow these steps:

### Step-by-Step Solution

1. **Rewrite the logarithms**:
We know that \( \log_{\sqrt{3}} x = \frac{\log_{3} x}{\log_{3} \sqrt{3}} = \frac{\log_{3} x}{\frac{1}{2}} = 2 \log_{3} x \). Thus, we can rewrite the equation:
\[ |2 \log_{3} x - 2| + |\log_{3} x - 2| = 2 \]

2. **Let \( y = \log_{3} x \)**:
Substituting \( y \) into the equation gives us:
\[ |2y - 2| + |y - 2| = 2 \]

3. **Analyze the absolute values**:
We will consider different cases based on the value of \( y \):

- **Case 1**: \( y \geq 2 \)
- Here, \( |2y - 2| = 2y - 2 \) and \( |y - 2| = y - 2 \).
- The equation becomes:
\[ (2y - 2) + (y - 2) = 2 \implies 3y - 4 = 2 \implies 3y = 6 \implies y = 2 \]
- Since \( y \geq 2 \), this solution is valid.

- **Case 2**: \( 1 \leq y < 2 \)
- Here, \( |2y - 2| = 2 - 2y \) and \( |y - 2| = 2 - y \).
- The equation becomes:
\[ (2 - 2y) + (2 - y) = 2 \implies 4 - 3y = 2 \implies 3y = 2 \implies y = \frac{2}{3} \]
- However, \( \frac{2}{3} < 1 \), so this solution is not valid.

- **Case 3**: \( y < 1 \)
- Here, \( |2y - 2| = 2 - 2y \) and \( |y - 2| = 2 - y \).
- The equation becomes:
\[ (2 - 2y) + (2 - y) = 2 \implies 4 - 3y = 2 \implies 3y = 2 \implies y = \frac{2}{3} \]
- Since \( y < 1 \) is satisfied, this solution is valid.

4. **Convert back to \( x \)**:
We have two valid solutions for \( y \):
- From Case 1: \( y = 2 \) leads to \( \log_{3} x = 2 \implies x = 3^2 = 9 \).
- From Case 3: \( y = \frac{2}{3} \) leads to \( \log_{3} x = \frac{2}{3} \implies x = 3^{\frac{2}{3}} = \sqrt[3]{9} \).

5. **Check the condition \( x > 2 \)**:
- For \( x = 9 \), \( 9 > 2 \) (valid).
- For \( x = \sqrt[3]{9} \approx 2.08 \), \( \sqrt[3]{9} > 2 \) (valid).

Thus, the solutions for \( x \) that satisfy the equation are \( x = 9 \) and \( x = \sqrt[3]{9} \).

### Final Answer
The values of \( x \) are \( 9 \) and \( \sqrt[3]{9} \).