Concavity of a term appearing in a log-likelihood function

Question

A came across following log-likelihood function with variables $x$ and $y$ with domain $x+y\leq0$ and some constant $c > 1$: \begin{equation} \log\mathcal{L}(x,y) = \log\left(1 + \log_c\left(1-\frac{1-c^{-1}}{e^{-x}+1}\right)+\log_c\left(1-\frac{1-c^{-1}}{e^{-y}+1}\right)\right) + \ldots \end{equation} I was able to show that all terms of this log-likelihood function are concave except for the first shown term. Hence, if this term was also concave, the whole log-likelihood function would be concave. Plotting this first term for various $c$ suggests that it is concave over the domain $x+y\leq 0$. See for example for $c=2$:

https://www.wolframalpha.com/input/?i=plot+ln%281%2Bln%281-%281-1%2F2%29%2F%28e%5E%28-x%29%2B1%29%29%2Fln%282%29%2Bln%281-%281-1%2F2%29%2F%28e%5E%28-y%29%2B1%29%29%2Fln%282%29%29+for+x+from+-3+to+3+and+for+y+from+-3+to+3

Does anyone know how to prove that it is really concave?