The domain of $\log x$ is the set of all $\underline{positive}$ reals, so the graph of $y=\log x$ clearly doesn't cut the $y$-axis. It therefore doesn't cut every line perpendicular to the $x$-axis. It does however cut the $x$-axis at $(1,0)$. In addition, if one examines the graph of $y=\log x$, one can clearly see that there are many circles centered at the origin that do not intersect the graph of $y=\log x$. Therefore the answer is $\boxed{\textbf{(C)}\ \text{Cuts the }x\text{-axis}}$.