`f(x)=sqrt((log(x-1))/(x^(2)-2x-8))`. Find the domain of f(x).

Answer

To find the domain of the function \( f(x) = \sqrt{\frac{\log(x-1)}{x^2 - 2x - 8}} \), we need to ensure that the expression inside the square root is non-negative. This means we need to solve the inequality:

\[ \frac{\log(x-1)}{x^2 - 2x - 8} \geq 0 \]

### Step 1: Identify conditions for the logarithm

The logarithm \( \log(x-1) \) is defined and non-negative when:

\[ x - 1 > 0 \implies x > 1 \]

### Step 2: Factor the denominator

Next, we need to factor the quadratic expression in the denominator:

\[ x^2 - 2x - 8 = (x - 4)(x + 2) \]

### Step 3: Determine the sign of the denominator

The expression \( (x - 4)(x + 2) \) will change sign at the points \( x = -2 \) and \( x = 4 \). We can analyze the sign of this expression in the intervals determined by these points:

- For \( x < -2 \): Both factors are negative, so the product is positive.
- For \( -2 < x < 4 \): The first factor is negative and the second is positive, so the product is negative.
- For \( x > 4 \): Both factors are positive, so the product is positive.

### Step 4: Set up cases for the inequality

Now we have two cases to consider based on the signs of the numerator and denominator:

**Case 1:** Both numerator and denominator are positive:
\[ \log(x-1) \geq 0 \quad \text{and} \quad (x - 4)(x + 2) > 0 \]

From the logarithm condition, we have \( x \geq 2 \). From the denominator, we have:

- \( x < -2 \) or \( x > 4 \)

Since \( x \geq 2 \), the only valid interval from this case is:

\[ x > 4 \]

**Case 2:** Both numerator and denominator are negative:
\[ \log(x-1) < 0 \quad \text{and} \quad (x - 4)(x + 2) < 0 \]

From the logarithm condition, we have \( 1 < x < 2 \). From the denominator, we have:

- \( -2 < x < 4 \)

The valid interval from this case is:

\[ 1 < x < 2 \]

### Step 5: Combine the intervals

Now we combine the intervals from both cases:

1. From Case 1: \( (4, \infty) \)
2. From Case 2: \( (1, 2) \)

Thus, the domain of \( f(x) \) is:

\[ (1, 2) \cup (4, \infty) \]

### Final Answer

The domain of the function \( f(x) \) is:

\[ \boxed{(1, 2) \cup (4, \infty)} \]