The number of roots of the equation $1+ \log_2 (1-x) = 2^{-x}$, is _______.

2

We observe that the curves $y = \log_2 (1-x)$ and $y=2^{-x}-1$ intersect at (0, 0) and (-1, 1) as shown in the following figure. So, the equation $1+ \log_2 (1-x)=2^{-x}$ has two solutions.