  The best polynomial approximation and Chebyshev approximation are both
important in numerical analysis. In tradition, the best approximation is
regarded as more better than the Chebyshev approximation, because it is usually
considered in the uniform norm. However, it not always superior to the latter
noticed by Trefethen \cite{Trefethen11sixmyths,Trefethen2020} for the algebraic
singularity function. Recently Wang \cite{Wang2021best} have proved it in
theory. In this paper, we find that for the functions with logarithmic
regularities, the pointwise errors of Chebyshev approximation are smaller than
the ones of the best approximations except only in the very narrow boundaries
at the same degree. The pointwise error for Chebyshev series, truncated at the
degree $n$ is $O(n^{-\kappa})$ ($\kappa = \min\{2\gamma+1, 2\delta + 1\}$), but
is worse by one power of $n$ in narrow boundary layer near the weak singular
endpoints. Theorems are given to explain this effect.
