Considering functions of the form: $$f\{p_1,p_2\}(x) = p_1(x)e^{p_2(x)} \text{ , where : } p_l(x) = \sum_{k=0}^N c_{lk}x^k$$
It seems to me, that if using a matrix representation using Toeplitz matrices of the same size as the maximal order polynomial used and replacing exponential with matrix exponential, all information is somehow preserved, fullfilling all algebra (at least of which I've tried so far), for example;
$$\log(P_1\exp(P_2)) - \log(P_1) - P_2 = 0$$
This is surprising to me, because I expect that the series expansion done by matrix exponential and logarithm must somehow lose precision or "information" in some sense, as the matrices can only store monomials of as high order as the largest polynomial.
Is there some way to explain this? Is there something important I am missing?