Hardy type derivations on fields of exponential logarithmic series

We consider the valued field K:=R((Γ)) of formal series (with real coefficients and monomials in a totally ordered multiplicative group Γ). We investigate how to endow K with a logarithm l, which satisfies some natural properties such as commuting with infinite products of monomials. We studied derivations on K (Kuhlmann and Matusinski, in press [KM10], ). Here, we investigate compatibility conditions between the logarithm and the derivation, i.e. when the logarithmic derivative is the derivative of the logarithm. We analyze sufficient conditions on a given derivation to construct a compatible logarithm via integration of logarithmic derivatives. In Kuhlmann (2000) [Kuh00], the first author described the exponential closure KEL of (K,l). Here we show how to extend such a log-compatible derivation on K to KEL.