One searches for a
$y$ satisfying the equation
$x^a=-a \log(1-y)$
In[16]:= Solve[x^a == -a Log[1 - y] && 0 < a < 1 && 1/2 < x < 1, y, Reals]
Out[16]= {{y -> ConditionalExpression[1 - E^(-(x^a/a)), 0 < a < 1 && 1/2 < x < 1]}}
that's trivial but allows to transform the equation into
$(1-x)\exp\left(-\frac{x^a}{a}\right)=\exp\left(-\frac{c}{a}\right)$. This will not be solved again
Solve[(1 - x) Exp[-x^a/a] == Exp[-c/a] && 1/2 < x < 1 && 0 < a < 1, x, Reals]
Solve::nsmet: This system cannot be solved with the methods available to Solve. >>
but can be expressed as
$(1-x)\exp\left(\frac{c-x^a}{a}\right)=1$.
It's interesting to see how different quality the corresponding ContourPlot3D
for the three equivalent equations are.
Any derivative of the above mentioned expression must vanish, but I do not see an usage of that fact.
But one could try to develop around
$a = 1$. At least for
$a=1$ a solution appears
In[36]:= Solve[(1 - x) Exp[c - x] == 1 , x]
During evaluation of In[36]:= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. >>
Out[36]= {{x -> 1 - ProductLog[E^(1 - c)]}}