  We provide direct measurements of the boundary layer properties in highly
turbulent Taylor-Couette flow up to $\text{Ta}=6.2 \times 10^{12}$ using
high-resolution particle image velocimetry (PIV). We find that the mean
azimuthal velocity profile at the inner and outer cylinder can be fitted by the
von K\'arm\'an log law $u^+ = \frac 1\kappa \ln y^+ +B$. The von K\'arm\'an
constant $\kappa$ is found to depend on the driving strength $\text{Ta}$ and
for large $\text{Ta}$ asymptotically approaches $\kappa \approx 0.40$. The
variance profiles of the local azimuthal velocity have a universal peak around
$y^+ \approx 12$ and collapse when rescaled with the driving velocity (and not
with the friction velocity), displaying a log-dependence of $y^+$ as also found
for channel and pipe flows [1,2].
