To analyze the problem based on the function $y = \log_3(x - 5) - 2$ and address each part:

(a) Transformations of the graph of $y = \log_3 x$:

1. Horizontal Shift: The function has $(x - 5)$ inside the logarithm, indicating a shift 5 units to the right.
2. Vertical Shift: The $-2$ outside the logarithm moves the graph down by 2 units.

These transformations should be applied to $y = \log_3 x$ to graph $y = \log_3(x - 5) - 2$.

(b) Domain and Range:

• Domain: Since $x - 5 > 0$, the domain is $x > 5$. In interval notation, this is $(5, \infty)$.
• Range: Since logarithmic functions are defined for all real values of $y$, the range is $(-\infty, \infty)$.

(c) Equation of the Asymptote:

• The vertical asymptote for the function occurs where the argument of the logarithm is zero. So, $x - 5 = 0 \Rightarrow x = 5$.

Therefore, the equation of the asymptote is $x = 5$.

Would you like further details on any specific part, or have additional questions?

1. What other transformations could be applied to a logarithmic function?
2. How does the base of the logarithm affect the shape of the graph?
3. What changes would occur if the vertical shift were positive instead of negative?
4. How would a reflection over the x-axis appear on the graph of a logarithmic function?
5. What happens to the domain if the logarithmic function is $\log_3(x + 5)$ instead?

Tip: Always identify the base of the logarithm when analyzing transformations, as it affects the rate of increase.