The unified description of fragmentation function evolution from large tosmall x which was developed for the vacuum in previous publications is nowgeneralized to the medium, and is studied for the case in which the completecontribution from the largest class of soft gluon logarithms, the doublelogarithms, are accounted for and with the fixed order part calculated toleading order. In this approach it proves possible to choose the remainingdegrees of freedom related to the medium such that the distribution of producedhadrons is suppressed at large momenta while the production of softradiation-induced charged hadrons at small momenta is enhanced, in agreementwith experiment. Just as for the vacuum, our approach does not require furtherassumptions concerning fragmentation and is more complete than previouscomputations of evolution in the medium.
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