Answer: f ( x ) = log ( x - 5 )
Explanation:
1) Take g(x) = log (x) as the parent function
2) The horizontal asymptote of g(x) is x = 0, and the y-intercept is (1,0).
3) Then, translating the asymptote to x = 5, and the y-intercept to (6,0), means that the graph of the parent function is being shifted 5 units to the right.
4) The rule is that the translation of the function g(x) a constant value to the right, say it is k, results in the function f(x) = g(x - k).
5) Therefore, the logarithmic function searched is f(x) = g(x - 5) = log (x - 5), and that is the answer.
6) You can prove that log (x - 5) meets the two conditions:
i) [tex] \lim_{x \to \\5+} log(x-5) = - \infty [/tex], which means x = 5 is a vertical asymptote
ii) f(6) = log (6 - 5) = log (1) = 0 ⇒ point (6,0) is the x-intercept
