The number of points on the real line where the function `f(x) = log_(|x^2-1|)|x-3|` is not defined is

Options

A. 4
B. 5
C. 6
D. 7

Explanation

To determine the number of points on the real line where the function \( f(x) = \log_{|x^2 - 1|} |x - 3| \) is not defined, we need to analyze the conditions under which the logarithm is defined.

### Step 1: Identify conditions for the logarithm to be defined
The logarithm \( \log_b(a) \) is defined under the following conditions:
1. \( a > 0 \)
2. \( b > 0 \) and \( b \neq 1 \)

In our case:
- \( a = |x - 3| \)
- \( b = |x^2 - 1| \)

### Step 2: Analyze the condition \( a > 0 \)
For \( |x - 3| > 0 \):
- This implies \( x - 3 \neq 0 \) or \( x \neq 3 \).

### Step 3: Analyze the condition \( b > 0 \)
For \( |x^2 - 1| > 0 \):
- This implies \( x^2 - 1 \neq 0 \) or \( x^2 \neq 1 \).
- Thus, \( x \neq 1 \) and \( x \neq -1 \).

### Step 4: Analyze the condition \( b \neq 1 \)
For \( |x^2 - 1| \neq 1 \):
- We have two cases to consider:
1. \( x^2 - 1 \neq 1 \) which simplifies to \( x^2 \neq 2 \) leading to \( x \neq \sqrt{2} \) and \( x \neq -\sqrt{2} \).
2. \( x^2 - 1 \neq -1 \) which simplifies to \( x^2 \neq 0 \) leading to \( x \neq 0 \).

### Step 5: Compile all points where \( f(x) \) is not defined
From the conditions derived:
- \( x \neq 3 \)
- \( x \neq 1 \)
- \( x \neq -1 \)
- \( x \neq \sqrt{2} \)
- \( x \neq -\sqrt{2} \)
- \( x \neq 0 \)

### Step 6: Count the points
We have the following points where the function is not defined:
1. \( x = 3 \)
2. \( x = 1 \)
3. \( x = -1 \)
4. \( x = \sqrt{2} \)
5. \( x = -\sqrt{2} \)
6. \( x = 0 \)

Thus, there are a total of **6 points** on the real line where the function \( f(x) \) is not defined.

### Final Answer
The number of points on the real line where the function \( f(x) \) is not defined is **6**.

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