  For renormalizable theories with a single coupling constant regularized by
higher derivatives we investigate the coefficients at powers of logarithms
present in the renormalization constants assuming that divergences are removed
by minimal subtractions of logarithms. According to this (HD+MSL)
renormalization prescription the renormalization constants include only powers
of $\ln\Lambda/\mu$, where $\Lambda$ and $\mu$ are the dimensionful
regularization parameter and the renormalization point, respectively. We
construct general explicit expressions for arbitrary coefficients at powers of
this logarithm present in the coupling constant renormalization and in the
field renormalization constant which relate them to the $\beta$-function and
(in the latter case) to the corresponding anomalous dimension. To check the
correctness, we compare the results with the explicit four-loop calculation
made earlier for ${\cal N}=1$ SQED and (for the supersymmetric case) rederive a
relation between the renormalization constants following from the NSVZ
equation.
