  We perform an all-order resurgence analysis of a quantum field theory
renormalon that contributes to an anomalous dimension in six-dimensional scalar
$\phi^3$ theory and is governed by a third-order nonlinear differential
equation. We augment the factorially divergent perturbative expansion
associated to the renormalon by asymptotic expansions to all instanton orders,
in a conjectured and well-tested formula. A distinctive feature of this
renormalon singularity is the appearance of logarithmic terms, starting at
second-instanton order in the trans-series. To highlight this and to illustrate
our methods, we also analyze the trans-series for a closely related
second-order nonlinear differential equation that exhibits a similarly resonant
structure but lacks logarithmic contributions.
