An object $X$ of a category $\mathbf{C}$ with finite limits is called exponentiable if the functor $-\times X:\mathbf{C}\rightarrow \mathbf{C}$ has a right adjoint. There are many characterizations of the exponentiable spaces in the category $\mathbf{Top}$ of topological spaces. Here, we study the exponentiable objects in the category $\mathbf{STop}$ of soft topological spaces which is a generalization of the category $\mathbf{Top}$. We investigate the exponentiability problem and give a characterization of exponentiable soft spaces. Also wegive the definition of exponential topology on the lattice of soft open sets of a soft space and present some characterizations of it.