With[{Pt = 0.5, [Sigma] = 0.1, S = 1},
   FindRoot[1/(6 [Sigma]^2 (-1 + w) w) S ((Log[Sqrt[3] [Sigma] + Pt] -
      Log[Pt + Sqrt[3] [Sigma] (1 - 2 w)]) Pt (Sqrt[3] [Sigma] + Pt) - (6 [Sigma]^2 +
       Sqrt[3] [Sigma] (Log[-Sqrt[3] [Sigma] + Pt] + Log[Sqrt[3] [Sigma] + Pt] - 2 Log[Pt + Sqrt[3] [Sigma] (1 - 2 w)]) Pt + (-Log[-Sqrt[
          3] [Sigma] + Pt] + Log[Sqrt[3] [Sigma] + Pt]) Pt^2) w + 6 [Sigma]^2 w^2) == 0, {w, 0.5}]]