  As an alternative to directly integrating their defining equations to find
the running coupling $a(\mu)$ and the running mass $m(\mu)$, we expand these
quantities in powers of $\ln\left(\frac{\mu}{\mu^\prime}\right)$ and their
boundary values $a(\mu^\prime)$ and $m(\mu^\prime)$. Renormalization group
summation is used to partially sum these logarithms. We consider this approach
using both the $\overline{MS}$ and 't Hooft renormalization schemes. We also
show how the couplings and masses in any two mass independent renormalization
schemes are related.
