Find the domain of the following functions : `f(x)=sqrt(log_10((log_10x)/(2(3-log_10x)))`

Explanation

To find the domain of the function \( f(x) = \sqrt{\log_{10}\left(\frac{\log_{10} x}{2(3 - \log_{10} x)}\right)} \), we need to ensure that the expression inside the square root is non-negative and that the logarithm is defined.

### Step-by-Step Solution

1. **Identify the Conditions for the Logarithm**:
The logarithm \( \log_{10} x \) is defined for \( x > 0 \). Therefore, we have:
\[ x > 0 \]

2. **Set the Argument of the Logarithm Greater than Zero**:
For the logarithm \( \log_{10}\left(\frac{\log_{10} x}{2(3 - \log_{10} x)}\right) \) to be defined, we need:
\[ \frac{\log_{10} x}{2(3 - \log_{10} x)} > 0 \]
This fraction is positive when both the numerator and denominator are either both positive or both negative.

3. **Analyze the Numerator**:
The numerator \( \log_{10} x > 0 \) implies:
\[ x > 10^0 \quad \Rightarrow \quad x > 1 \]

4. **Analyze the Denominator**:
The denominator \( 2(3 - \log_{10} x) > 0 \) implies:
\[ 3 - \log_{10} x > 0 \quad \Rightarrow \quad \log_{10} x < 3 \quad \Rightarrow \quad x < 10^3 \quad \Rightarrow \quad x < 1000 \]

5. **Combine the Conditions**:
From the analysis, we have:
- \( x > 1 \)
- \( x < 1000 \)

Therefore, combining these inequalities gives:
\[ 1 < x < 1000 \]

6. **Conclusion**:
The domain of the function \( f(x) \) is:
\[ (1, 1000) \]

### Final Domain
Thus, the domain of the function \( f(x) = \sqrt{\log_{10}\left(\frac{\log_{10} x}{2(3 - \log_{10} x)}\right)} \) is:
\[ \boxed{(1, 1000)} \]