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The problem asks to find the domain and vertical asymptote for the function \( f(x) = \log_3(x + 7) \).

### 1. **Domain of the function**:
The logarithmic function \( \log_3(x + 7) \) is only defined when its argument is positive, i.e., when:

\[
x + 7 > 0
\]

Solving for \( x \):

\[
x > -7
\]

Thus, the domain is:

\[
\text{Dom } f = (-7, \infty)
\]

### 2. **Vertical asymptote**:
The vertical asymptote occurs where the argument of the logarithm approaches zero because logarithmic functions tend to \( -\infty \) as the argument approaches zero from the right. So, set the argument equal to zero:

\[
x + 7 = 0 \quad \Rightarrow \quad x = -7
\]

Thus, there is a vertical asymptote at \( x = -7 \).

### Conclusion:
- The domain is \( (-7, \infty) \).
- The vertical asymptote is at \( x = -7 \).

The correct answer from the options provided is:

\[
\boxed{\text{Dom } f = (-7, \infty), \, x = -7}
\]

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Do you need more details or have any questions about this? Here are some additional questions for practice:

1. What is the domain of \( f(x) = \log_5(x - 3) \)?
2. How do you find the vertical asymptote of a general logarithmic function?
3. What happens to the graph of \( f(x) = \log_b(x) \) when \( b \) changes?
4. How does adding or subtracting a constant inside the logarithm affect its domain?
5. What is the domain of \( f(x) = \log(x^2 - 1) \)?

**Tip:** Always check that the argument of a logarithmic function is positive to determine its domain!

Dada la función f(x) = log_3(x + 7) hallar el dominio y la asíntota respectivamente:

Learn how to find the domain and vertical asymptote of the logarithmic function f(x) = log_3(x + 7). This problem demonstrates the concepts of logarithmic function domain determination and vertical asymptote location. Suitable for students in Grades 9-11.

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