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?

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.