  We report results for the temperature profiles of turbulent Rayleigh-B\'enard
convection (RBC) in the interior of a cylindrical sample of aspect ratio
$\Gamma \equiv D/L = 0.50$ ($D$ and $L$ are the diameter and height
respectively). Results from experiment over the Rayleigh number range $4\times
10^{12} \alt Ra \alt 10^{15}$ for a Prandtl number $\Pra \simeq 0.8$ and from
direct numerical simulation (DNS) at $Ra = 2 \times 10^{12}$ for $\Pra = 0.7$
are presented. We find that the temperature varies as $A*ln(z/L) + B$ where $z$
is the distance from the bottom or top plate. This is the case in the classical
as well as in the ultimate state of RBC. From DNS we find that $A$ in the
classical state decreases in the radial direction as the distance from the side
wall increases and becomes small near the sample center.
