User the function $f(x)=\log _{5}(x-3)-2$ to answer the following: (a) Determine the equation of the asymptote of $f(x)$. (b) Determine the domain of $f(x)$ in interval notation. (c) Determine the range of $f(x)$ in interval notation.

Step 1 ：Set the argument of the logarithm equal to zero: \(x-3=0\)

Step 2 ：Solve for \(x\) to get \(x=3\)

Step 3 ：\(\boxed{x=3}\) is the equation of the asymptote of \(f(x)\)

Step 4 ：Set \(x-3>0\) to find the domain of \(f(x)\)

Step 5 ：Solve for \(x\) to get \(x>3\)

Step 6 ：The domain of \(f(x)\) in interval notation is \(\boxed{(3, \infty)}\)

Step 7 ：Since the base of the logarithm is greater than 1, the function is increasing for all values in its domain

Step 8 ：Therefore, the function will take on all real number values

Step 9 ：The range of \(f(x)\) in interval notation is \(\boxed{(-\infty, \infty)}\)