In a recent work published in this journal [JNT \textbf{129}, 2154 (2009)],
it has been argued that the numbers $\log{\Gamma(x)} + \log{\Gamma(1-x)}$, $x$
being a rational number between 0 and 1, are transcendental with at most
\emph{one} possible exception, but the proof presented there is
\emph{incorrect}. Here in this paper, I point out the mistake committed in that
proof and I present a theorem that establishes the transcendence of those
numbers, with at most \emph{two} possible exceptions. This yields a criteria
for the algebraicity of $\log{\pi}$, a number that presently is not known even
to be irrational. I also show that each pair $\{\log{[\pi/\sin(\pi x)]},
\log{[\pi/\sin(\pi y)]}\}$ contains at least one transcendental number, e.g.
$\{\log{\pi}, \log{(2 \pi)} \}$. With respect to this pair, I show that if $
\log{(k \pi)}$ is algebraic for some non-zero algebraic $k$ then the product
$\pi e$, another number whose irrationality is not proved, has to be
transcendental.