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., $x > 0$).