  Invariance of the effective action under changes of the renormalization scale
$\mu$ leads to relations between those (presumably calculated) terms
independent of $\mu$ at a given order of perturbation theory and those higher
order terms dependent on logarithms of $\mu$. This relationship leads to
differential equations for a sequence of functions, the solutions of which give
closed form expressions for the sum of all leading logs, next to leading logs
and subsequent subleading logarithmic contributions to the effective action.
The renormalization group is thus shown to provide information about a model
beyond the scale dependence of the model's couplings and masses. This procedure
is illustrated using the $\phi_6^3$ model and Yang-Mills theory. In the latter
instance, it is also shown by using a modified summation procedure that the
$\mu$ dependence of the effective action resides solely in a multiplicative
factor of $g^2 (\mu)$ (the running coupling). This approach is also shown to
lead to a novel expansion for the running coupling in terms of the one-loop
coupling that does not require an order-by-order redefinition of the scale
factor $\Lambda_{QCD}$. Finally, logarithmic contributions of the instanton
size to the effective action of an SU(2) gauge theory are summed, allowing a
determination of the asymptotic dependence on the instanton size $\rho$ as
$\rho$ goes to infinity to all orders in the SU(2) coupling constant.
