To find the domain of the function $f(x) = \log_7(4x + 8)$, we need to ensure that the argument of the logarithm is positive, because the logarithmic function is only defined for positive values of its argument.

Step-by-Step Solution:

1. The argument inside the logarithm is $4x + 8$. To ensure this is positive, we set up the inequality: $4x + 8 > 0$

2. Solve the inequality for $x$: $4x > -8$ $x > -2$

Thus, the domain of $f(x) = \log_7(4x + 8)$ is $x > -2$.

Final Answer:

The domain of the function is: $(-2, \infty)$

Would you like further details on this, or do you have any questions?

Here are five related questions:

1. How do you find the domain of other logarithmic functions?
2. What happens if the base of the logarithm changes?
3. How does shifting the function horizontally affect the domain?
4. How do you solve logarithmic inequalities?
5. Can logarithmic functions ever have a domain that includes negative numbers?

Tip: When finding the domain of a logarithmic function, always ensure the argument inside the log is strictly positive.