  We present new results on the analytic eccentricity dependence of several
sequences of gravitational wave flux terms at high post-Newtonian (PN) order
for extreme-mass-ratio inspirals. These sequences are the leading logarithms,
which appear at PN orders $x^{3k} \log^k(x)$ and $x^{3k+3/2} \log^k(x)$ for
integers $k\ge 0$ ($x$ a PN compactness parameter), and the subleading
logarithms, which appear at orders $x^{3k} \log^{k-1}(x)$ and $x^{3k+3/2}
\log^{k-1}(x)$ ($k\ge 1$), in both the energy and angular momentum radiated to
infinity. For the energy flux leading logarithms, we show that to arbitrarily
high PN order their eccentricity dependence is determined by particular sums
over the function $g(n,e_t)$, derived from the Newtonian mass quadrupole
moment, that normally gives the spectral content of the Peters-Mathews flux as
a function of radial harmonic $n$. An analogous power spectrum
$\tilde{g}(n,e_t)$ determines the leading logarithms of the angular momentum
flux. For subleading logs, the quadrupole power spectra are again shown to play
a role, providing a distinguishable part of the eccentricity dependence of
these flux terms to high PN order. With the quadrupole contribution understood,
the remaining analytic eccentricity dependence of the subleading logs can in
principle be determined more easily using black hole perturbation theory. We
show this procedure in action, deriving the complete analytic structure of the
$x^6 \log(x)$ subleading-log term and an analytic expansion of the $x^{9/2}$
subleading log to high order in a power series in eccentricity. We discuss how
these methods might be extended to other sequences of terms in the PN expansion
involving logarithms.
