Complete set of real values of x for which `log_((2x-3))(x^(2)-5x-6)` is defined is :

Options

A. `((3)/(2),oo)`
B. `(6, oo)`
C. `((3)/(2),6)`
D. `((3)/(2),2) cup (2, oo)`

Answer

To solve the problem of finding the complete set of real values of \( x \) for which \( \log_{(2x-3)}(x^2 - 5x - 6) \) is defined, we need to satisfy several conditions based on the properties of logarithms. Let's break it down step by step.

### Step 1: Identify the conditions for the logarithm to be defined
For the logarithm \( \log_b(a) \) to be defined, the following conditions must be satisfied:
1. \( b > 0 \) and \( b \neq 1 \)
2. \( a > 0 \)

In our case:
- \( a = x^2 - 5x - 6 \)
- \( b = 2x - 3 \)

### Step 2: Set up the inequalities based on the conditions

#### Condition 1: \( b > 0 \)
\[ 2x - 3 > 0 \implies 2x > 3 \implies x > \frac{3}{2} \]

#### Condition 2: \( b \neq 1 \)
\[ 2x - 3 \neq 1 \implies 2x \neq 4 \implies x \neq 2 \]

#### Condition 3: \( a > 0 \)
We need to solve the inequality:
\[ x^2 - 5x - 6 > 0 \]
First, we can factor the quadratic:
\[ x^2 - 5x - 6 = (x - 6)(x + 1) \]
Now we find the roots:
\[ x - 6 = 0 \implies x = 6 \]
\[ x + 1 = 0 \implies x = -1 \]
Next, we analyze the sign of the product \( (x - 6)(x + 1) \):
- The critical points are \( x = -1 \) and \( x = 6 \).
- Testing intervals:
- For \( x < -1 \): both factors are negative, so the product is positive.
- For \( -1 < x < 6 \): one factor is negative and the other is positive, so the product is negative.
- For \( x > 6 \): both factors are positive, so the product is positive.

Thus, the solution to \( x^2 - 5x - 6 > 0 \) is:
\[ x \in (-\infty, -1) \cup (6, \infty) \]

### Step 3: Combine all conditions
Now we combine all the conditions we derived:
1. \( x > \frac{3}{2} \)
2. \( x \neq 2 \)
3. \( x \in (-\infty, -1) \cup (6, \infty) \)

From the conditions, we see:
- The interval \( (-\infty, -1) \) does not intersect with \( x > \frac{3}{2} \).
- The interval \( (6, \infty) \) does intersect with \( x > \frac{3}{2} \).

### Final Solution
Thus, the complete set of real values of \( x \) for which \( \log_{(2x-3)}(x^2 - 5x - 6) \) is defined is:
\[ \boxed{(6, \infty)} \]