Do logarthmic functions have horizontal asymptotes?

No they don't. They just stretch for a very long ways horizontally without much increase vertically because the output of the function is the exponent of the input. For example, f(x) = log x when x = 1000, f(x) = 3 because 10^3 = 1000 (10 being the base of common log). Therefore, when you increase x substantially, there is only a small increase in y.