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