Think Domain of of y = e^x and Range of y = ln(x).
does this make sense:
Since they are inverse functions, every (x,y) pair of ln(x) corresponds to a pair (y,x) of e^x. If ln(x) approaches a horizontal asymptote, that means that y has a limit. That means there is a limit on the x values of f(x). This is ridiculous, so obviously there is no horizontal asymptote for ln(x).