  We give a precise connection between combinatorial Dyson-Schwinger equations
and log-expansions for Green's functions in quantum field theory. The latter
are triangular power series in the coupling constant $\alpha$ and a logarithmic
energy scale $L$ --- a reordering of terms as $G(\alpha,L) = 1 \pm \sum_{j \geq
0} \alpha^j H_j(\alpha L)$ is the corresponding log-expansion. In a first part
of this paper, we derive the leading-log order $H_0$ and the
next-to$^{(j)}$-leading log orders $H_j$ from the Callan-Symanzik equation. In
particular, $H_j$ only depends on the $(j+1)$-loop $\beta$-function and
anomalous dimensions. For the photon propagator Green's function in quantum
electrodynamics (and in a toy model, where all Feynman graphs with vertex
sub-divergences are neglected), our formulas reproduce the known expressions
for the next-to-next-to-leading log approximation in the literature. In a
second part of this work, we review the connection between the Callan-Symanzik
equation and Dyson-Schwinger equations, i.e. fixed-point relations for the
Green's functions. Combining the arguments, our work provides a derivation of
the log-expansions for Green's functions from the corresponding Dyson-Schwinger
equations.
