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To find the domain of the function \( f(x) = \ln(x^2 - 9) \), we need to ensure that the expression inside the natural logarithm, \( x^2 - 9 \), is greater than zero because the logarithm is only defined for positive numbers.

### Step-by-Step Solution:

1. **Set the inside expression greater than zero:**
   \[
   x^2 - 9 > 0
   \]

2. **Solve for \( x \):**

   \[
   x^2 > 9
   \]

   Taking the square root of both sides:

   \[
   x > 3 \quad \text{or} \quad x < -3
   \]

3. **Determine the domain:**

   The domain of the function is all \( x \) values such that \( x > 3 \) or \( x < -3 \).

   In interval notation, this is:

   \[
   (-\infty, -3) \cup (3, \infty)
   \]

### Conclusion:

The domain of the function \( f(x) = \ln(x^2 - 9) \) is \( (-\infty, -3) \cup (3, \infty) \).

Would you like more details, or do you have any questions about this solution?

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Here are 5 related questions to expand your understanding:

1. What is the domain of the function \( f(x) = \ln(x - 2) \)?
2. How do you find the domain of a function involving a square root, like \( g(x) = \sqrt{x - 4} \)?
3. What happens to the domain of a logarithmic function if the inside expression is a quadratic trinomial?
4. How do you graph a logarithmic function such as \( f(x) = \ln(x^2 - 9) \)?
5. Can a logarithmic function have a negative inside expression, and why or why not?

**Tip:** Remember, the domain of a logarithmic function is determined by the requirement that its argument must be positive.

Find the domain of the function f(x) = ln(x^2 - 9).

Discover how to find the domain of the logarithmic function f(x) = ln(x^2 - 9) with this comprehensive guide. This problem involves key algebraic concepts, including solving quadratic inequalities, and is suitable for students in Grades 10-12.

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