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 \( f(x) = \log_3\left(1 - \log_6(x^2 - 7x + 16)\right) \), we need to ensure that the expression inside the logarithm is positive. Let's break it down step by step.

### Step 1: Set up the inequality
We need to ensure that:
\[ 1 - \log_6(x^2 - 7x + 16) > 0 \]
This implies:
\[ \log_6(x^2 - 7x + 16) < 1 \]

### Step 2: Convert the logarithmic inequality
We can rewrite the inequality:
\[ x^2 - 7x + 16 < 6^1 \]
This simplifies to:
\[ x^2 - 7x + 16 < 6 \]
or
\[ x^2 - 7x + 10 < 0 \]

### Step 3: Factor the quadratic
Now, we need to factor the quadratic expression:
\[ x^2 - 7x + 10 = (x - 5)(x - 2) \]
Thus, we have:
\[ (x - 5)(x - 2) < 0 \]

### Step 4: Determine the intervals
To find the intervals where this product is negative, we can use a sign chart. The roots of the equation are \( x = 2 \) and \( x = 5 \). We test the intervals:
1. For \( x < 2 \): both factors are negative, so the product is positive.
2. For \( 2 < x < 5 \): one factor is negative and the other is positive, so the product is negative.
3. For \( x > 5 \): both factors are positive, so the product is positive.

Thus, the solution to the inequality \( (x - 5)(x - 2) < 0 \) is:
\[ 2 < x < 5 \]

### Step 5: Check the quadratic expression
Next, we need to ensure that the expression \( x^2 - 7x + 16 \) is positive for all real \( x \). We can check this by calculating the discriminant:
\[ D = b^2 - 4ac = (-7)^2 - 4 \cdot 1 \cdot 16 = 49 - 64 = -15 \]
Since the discriminant is negative, the quadratic \( x^2 - 7x + 16 \) is always positive.

### Final Domain
Thus, the domain of the function \( f(x) \) is:
\[ \boxed{(2, 5)} \]