Step 1 :The function given is \(f(x)=\log _{5}(x-6)\).
Step 2 :The domain of a logarithmic function is the set of all real numbers for which the argument of the logarithm is greater than zero. This is because the logarithm of a negative number or zero is undefined.
Step 3 :Therefore, we need to solve the inequality \(x-6 > 0\) to find the domain of the function.
Step 4 :The solution to the inequality \(x - 6 > 0\) is \(x > 6\).
Step 5 :This means that the domain of the function \(f(x) = \log_{5}(x - 6)\) is all real numbers greater than 6.
Step 6 :\(\boxed{\text{The domain of the function is }(6, \infty)}\).