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We combine different techniques to extract information about the logarithmiccontributions to the two-body conservative dynamics within the post-Newtonian(PN) approximation of General Relativity. The logarithms come from theconservative part of non linear gravitational-wave tails and their iterations.Explicit, original expressions are found for conservative dynamics logarithmictail terms up to 6PN order by adopting both traditional PN calculations andeffective field theory (EFT) methods. We also determine all logarithmic termsat 7PN order, fixing a sub-leading logarithm from a tail-of-tail-of-tailprocess by comparison with self-force (SF) results. Moreover, we userenormalization group techniques to obtain the leading logarithmic terms togeneric power $n$, appearing at $(3n+1)$PN order, and we resum the infiniteseries in a closed form. Half-integer PN orders enter the conservative dynamicsstarting at 5.5PN, but they do not generate logarithmic contributions up tonext-to-next-to-leading order included. We nevertheless present theircontribution at leading order in the small mass ratio limit.

Logarithmic tail contributions to the energy function of circular compact binaries