find the domain of $$ \log_4\left(x^2-64\right) $$

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.

Learn how to find the domain of the logarithmic function log_4(x^2 - 64) with a detailed step-by-step solution. This problem involves solving inequalities, understanding logarithmic functions, and working with quadratic expressions. Suitable for students in Grades 9-11.

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