Find the domain of each of the following functions: `f(x)=sqrt((log_(2)(x-2))/(log_(1//2)(3x-1)))`

Explanation

To find the domain of the function \( f(x) = \sqrt{\frac{\log_2(x-2)}{\log_{1/2}(3x-1)}} \), we need to ensure that the expression inside the square root is defined and non-negative. This means we need to consider the conditions for both the logarithmic functions and the square root.

### Step 1: Conditions for the logarithmic functions

1. **For \( \log_2(x-2) \)**:
- The argument of the logarithm must be greater than zero:
\[ x - 2 > 0 \implies x > 2 \]

2. **For \( \log_{1/2}(3x-1) \)**:
- The argument of the logarithm must also be greater than zero:
\[ 3x - 1 > 0 \implies x > \frac{1}{3} \]

### Step 2: Conditions for the square root

The expression inside the square root must be non-negative:
\[ \frac{\log_2(x-2)}{\log_{1/2}(3x-1)} \geq 0 \]

Since \( \log_{1/2}(y) \) is negative for \( y > 1 \) (because the base is less than 1), we need to find when \( 3x - 1 < 1 \):
\[ 3x - 1 < 1 \implies 3x < 2 \implies x < \frac{2}{3} \]

### Step 3: Combine the conditions

Now we combine the inequalities:
1. From \( x > 2 \)
2. From \( x > \frac{1}{3} \)
3. From \( x < \frac{2}{3} \)

The first condition \( x > 2 \) contradicts the third condition \( x < \frac{2}{3} \). Therefore, there are no values of \( x \) that satisfy all conditions simultaneously.

### Conclusion

The domain of the function \( f(x) \) is empty, meaning there are no valid inputs for which the function is defined.