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The given function is \( f(x) = \log_7(12 - 4x) \).

To find the domain of the logarithmic function, we need to determine when the argument inside the logarithm is positive. That is:

\[
12 - 4x > 0
\]

### Step-by-step solution:

1. Solve for \(x\):
   \[
   12 > 4x
   \]
   \[
   \frac{12}{4} > x
   \]
   \[
   3 > x \quad \text{or} \quad x < 3
   \]

2. Therefore, the domain of the function is all values of \(x\) such that \(x < 3\).

### Final Answer:
The domain in interval notation is:

\[
(-\infty, 3)
\]

Would you like any further details or explanations?

Here are 5 related questions to deepen your understanding:
1. What happens to the domain of a logarithmic function if the base of the logarithm changes?
2. How would the domain change if the function was \( f(x) = \log_7(4x - 12) \)?
3. Can a logarithmic function have a zero or negative argument, and why not?
4. How do you determine the range of a logarithmic function like this one?
5. What is the general form of the domain for any logarithmic function \( \log_b(g(x)) \)?

**Tip**: Always remember, for any logarithmic function \( \log_b(x) \), the argument \(x\) must be strictly positive (i.e., \(x > 0\)).

Find the domain of the function f(x) = log_7(12 - 4x).

Learn how to find the domain of the logarithmic function f(x) = log_7(12 - 4x). This problem explains the necessary condition for the argument inside the logarithm to be positive and the process of solving the inequality to determine the domain. Suitable for high school students (Grades 9-12).

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