To find the domain of the function

$f(x) = \log_4(x^2 - 64),$

we need to ensure that the argument of the logarithm is positive. Recall that the logarithm of a number is defined only when its argument is positive. Therefore, we require:

$x^2 - 64 > 0.$

Let's solve this inequality step-by-step:

Step 1: Solve the quadratic inequality

The inequality $x^2 - 64 > 0$ can be rewritten as:

$x^2 > 64.$

Taking the square root of both sides, we get:

$|x| > 8.$

This absolute value inequality gives us two cases:

1. $x > 8$
2. $x < -8$

Step 2: Combine the intervals

The solution to $|x| > 8$ consists of two separate intervals:

$x \in (-\infty, -8) \cup (8, \infty).$

Conclusion

Thus, the domain of the function $f(x) = \log_4(x^2 - 64)$ is:

$\boxed{(-\infty, -8) \cup (8, \infty)}.$

This means the function is defined for all values of $x$ that are either less than $-8$ or greater than $8$.

Would you like a more detailed explanation, or do you have any questions?

Related Questions to Explore:

1. How do you find the domain of a square root function?
2. What is the range of the function $f(x) = \log_4(x^2 - 64)$?
3. How do you solve absolute value inequalities in general?
4. What are the properties of logarithmic functions?
5. How do you differentiate logarithmic functions?

Tip: When solving logarithmic functions, always start by ensuring the argument inside the logarithm is positive.