The domain of the function `f(x)=log_(3)[1-log_(6)(x^(2)-7x+16)]` is

Options

A. (2, 5)
B. `(oo, 5)`
C. `[2, oo)`
D. `[2, 5]`

Answer

To find the domain of the function \( f(x) = \log_3(1 - \log_6(x^2 - 7x + 16)) \), we need to ensure that the arguments of the logarithmic functions are valid. This involves two main conditions:

1. The expression inside the logarithm \( x^2 - 7x + 16 \) must be greater than 0.
2. The expression \( 1 - \log_6(x^2 - 7x + 16) \) must be greater than 0.

Let's solve these conditions step by step.

### Step 1: Solve \( x^2 - 7x + 16 > 0 \)

To determine when \( x^2 - 7x + 16 \) is greater than 0, we can analyze the quadratic expression.

1. **Calculate the discriminant**:
\[ D = b^2 - 4ac = (-7)^2 - 4 \cdot 1 \cdot 16 = 49 - 64 = -15 \]
Since the discriminant is negative, the quadratic does not have real roots and is always positive (as the coefficient of \( x^2 \) is positive).

Thus, \( x^2 - 7x + 16 > 0 \) for all \( x \in \mathbb{R} \).

### Step 2: Solve \( 1 - \log_6(x^2 - 7x + 16) > 0 \)

This condition can be rewritten as:
\[ \log_6(x^2 - 7x + 16) < 1 \]

2. **Convert the logarithmic inequality**:
\[ x^2 - 7x + 16 < 6^1 = 6 \]

3. **Rearranging the inequality**:
\[ x^2 - 7x + 10 < 0 \]

4. **Factoring the quadratic**:
\[ x^2 - 7x + 10 = (x - 5)(x - 2) \]

5. **Finding the critical points**:
The critical points are \( x = 2 \) and \( x = 5 \).

6. **Using the wavy curve method**:
We analyze the sign of \( (x - 5)(x - 2) \):
- For \( x < 2 \): both factors are negative, product is positive.
- For \( 2 < x < 5 \): one factor is negative, the other is positive, product is negative.
- For \( x > 5 \): both factors are positive, product is positive.

Thus, \( (x - 5)(x - 2) < 0 \) for \( x \in (2, 5) \).

### Step 3: Combine the results

From Step 1, we found that \( x^2 - 7x + 16 > 0 \) for all \( x \in \mathbb{R} \).

From Step 2, we found that \( 1 - \log_6(x^2 - 7x + 16) > 0 \) is satisfied for \( x \in (2, 5) \).

### Conclusion

The domain of the function \( f(x) \) is:
\[ \boxed{(2, 5)} \]