To address the problem in the image:

The problem involves transforming the function $f(x) = \log_3{x}$ to $g(x) = \log_3{(x+5)}$.

Step 1: Analyze the transformation

The transformation involves changing $x$ to $x+5$. This indicates a horizontal shift of the graph of $f(x) = \log_3{x}$ by 5 units to the left, because adding $+5$ to $x$ shifts the graph left.

Step 2: Vertical asymptote

The vertical asymptote of a logarithmic function occurs where the argument of the logarithm becomes zero. For $g(x) = \log_3{(x+5)}$: $x+5 = 0 \implies x = -5$ Thus, the vertical asymptote is at $x = -5$.

Step 3: Domain and range

• Domain: The argument of the logarithm, $x+5$, must be positive: $x+5 > 0 \implies x > -5$ So, the domain is $(-5, \infty)$.

• Range: The range of any logarithmic function is $(-\infty, \infty)$.

Correct answers:

1. The transformation is 5 units to the left.
2. The vertical asymptote of $g(x) = \log_3{(x+5)}$ is $x = -5$.

Would you like detailed graphs or further explanations?

Here are 5 related questions for further exploration:

1. What is the effect of a vertical shift on the graph of a logarithmic function?
2. How does changing the base of a logarithm affect the graph?
3. Can you determine the horizontal asymptote for any transformed logarithmic function?
4. How do we find the inverse of $g(x) = \log_3{(x+5)}$?
5. How does the domain of $\log_3{(x+5)}$ change if the transformation is $\log_3{(x-5)}$?

Tip: Always rewrite logarithmic functions in terms of their transformations to quickly identify shifts and asymptotes.