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I've thought of this and it doesn't help me answer the second part of the question; This is because $v_L(Log(1+\pi_L^j))$ is difficult to deduce if $v_L(\pi_L^j) = v_L(1-r)$ for some $r\in\mu_{p^\infty}.$ (in all other cases this value can be calculated using the newton copolygon). This makes finding gererators for Log(1+m_K) hard for an arbitary extension L.
