Sure, you can apply FixedPoint[] as well as FixedPointList[]:
f[x_] := Log[x] + x - 0.5
y[x_] := x - \[Alpha] f[x]/f'[x]
Table[Print["\[Alpha] = ", \[Alpha], " ",
FixedPoint[# + (4 (y[#] - #) f[#])/(3 f[#] - 2 f[y[#]]) &,
0.75]], {\[Alpha], 0, 1, 0.01}]