The problem involves analyzing transformations of the logarithmic function $f(x) = \log_3(x)$ into $g(x) = \log_3(x+5)$. Let's break it down step by step:

1. Transformation:

   • The function $g(x) = \log_3(x+5)$ represents a horizontal shift of $f(x) = \log_3(x)$.
   • Specifically, adding $+5$ inside the logarithmic argument shifts the graph 5 units to the left. This matches option A in the provided choices.

2. Vertical Asymptote:

   • The original vertical asymptote of $f(x) = \log_3(x)$ is at $x = 0$.
   • After shifting the graph 5 units to the left, the vertical asymptote for $g(x) = \log_3(x+5)$ becomes $x = -5$.

3. Domain and Range:

   • The domain of $g(x) = \log_3(x+5)$ is $x > -5$, because the argument of the logarithm $(x+5) > 0$.
   • The range remains the same as $f(x)$, which is all real numbers $(-\infty, \infty)$.

Would you like further explanations or a graphing tool walkthrough?

Here are 5 related questions to deepen your understanding:

1. How does the base $3$ of the logarithm affect the graph's steepness?
2. What happens to the graph if $g(x) = \log_3(x-5)$?
3. How would the domain change if the argument was $\log_3(-x+5)$?
4. Can you describe the transformation if $g(x) = 2 \cdot \log_3(x+5)$?
5. How would the graph look if it were vertically reflected?

Tip: Always analyze transformations in terms of shifts, stretches, and reflections systematically!