This post is a recording of a proof that the logarithm function can not be expressed as the ratio of two polymials. Actually this can be easily inducted by judging the domain of the function.
Proof:Suppose that can be expressed as .We may assume that and . As we we have already known the rule of that , we have holds for every ,that is Now consider the first term of polynomial , that is either or or (only when or ). Then we get a contradiction by .