I tried:
Maximize[{log (1 + x) + b log (2 - x), 1 >= x >= 0}, x]
and got plenty of output (conditional of course, as Neslon said must be).
Mathematica gives the maximum values with conditions and then the x coordinates which have them - together in a list.
x = 2-b / 1+b is an extraneous formula to the solution.
I suggest you think about whether the 3 solutions Mathematica gives can be put into that form using algebra and the conditions.
