The domain of the function `y=sqrt(log_10(log_10x)-log_10(4-log_10x)-log_10 3 quad` is

Options

A. `(10^(3),10^(4))`
B. `[10^(3),10^(4)]`
C. `[10^(3),10^(4))`
D. `(10^(3),10^(4)]`

Answer

To find the domain of the function \( y = \sqrt{\log_{10}(\log_{10} x) - \log_{10}(4 - \log_{10} x) - \log_{10} 3} \), we need to ensure that the expression inside the square root is non-negative.

### Step-by-Step Solution:

1. **Set Up the Inequality**:
We want:
\[ \log_{10}(\log_{10} x) - \log_{10}(4 - \log_{10} x) - \log_{10} 3 \geq 0 \]

2. **Combine Logarithms**:
Using the property of logarithms that states \( \log_a b - \log_a c = \log_a \left(\frac{b}{c}\right) \), we can rewrite the inequality as:
\[ \log_{10} \left( \frac{\log_{10} x}{(4 - \log_{10} x) \cdot 3} \right) \geq 0 \]

3. **Exponentiate Both Sides**:
This implies:
\[ \frac{\log_{10} x}{(4 - \log_{10} x) \cdot 3} \geq 1 \]

4. **Rearranging the Inequality**:
Multiplying both sides by \( (4 - \log_{10} x) \cdot 3 \) (which we need to ensure is positive):
\[ \log_{10} x \geq (4 - \log_{10} x) \cdot 3 \]
Expanding the right-hand side gives:
\[ \log_{10} x \geq 12 - 3\log_{10} x \]

5. **Combine Like Terms**:
Adding \( 3\log_{10} x \) to both sides:
\[ 4\log_{10} x \geq 12 \]

6. **Solve for \( \log_{10} x \)**:
Dividing both sides by 4:
\[ \log_{10} x \geq 3 \]

7. **Exponentiate Again**:
This means:
\[ x \geq 10^3 = 1000 \]

8. **Check the Denominator**:
We also need to ensure that \( 4 - \log_{10} x > 0 \):
\[ \log_{10} x < 4 \implies x < 10^4 = 10000 \]

9. **Combine the Results**:
From the inequalities, we have:
\[ 1000 \leq x < 10000 \]

### Conclusion:
The domain of the function is:
\[ [1000, 10000) \]