  In cuprate superconductors, high magnetic fields have been used extensively
to suppress superconductivity and expose the underlying normal state. Early
measurements revealed insulating-like behavior in underdoped material versus
temperature $T$, in which resistivity increases on cooling with a puzzling
$\log(1/T)$ form. We instead use microwave measurements of flux-flow
resistivity in YBa$_2$Cu$_3$O$_{6+y}$ and Tl$_2$Ba$_2$CuO$_{6+\delta}$ to study
charge transport deep inside the superconducting phase, in the low temperature
and low field regime. Here, the transition from metallic low-temperature
resistivity ($d\rho/dT>0$) to a $\log(1/T)$ upturn persists throughout the
superconducting doping range, including a regime at high carrier dopings in
which the field-revealed normal-state resistivity is Fermi-liquid-like. The
$\log(1/T)$ form is thus likely a signature of $d$-wave superconducting order,
and the field-revealed normal state's $\log(1/T)$ resistivity may indicate the
free-flux-flow regime of a phase-disordered $d$-wave superconductor.
