We characterize the support of the law of the exponential functional $\int_0^\infty e^{-ξ_{s-}} \, dη_s$ of two one-dimensional independent Lévy processes $ξ$ and $η$. Further, we study the range of the mapping $Φ_ξ$ for a fixed Lévy process $ξ$, which maps the law of $η_1$ to the law of the corresponding exponential functional $\int_0^\infty e^{-ξ_{s-}} \, dη_s$. It is shown that the range of this mapping is closed under weak convergence and in the special case of positive distributions several characterizations of laws in the range are given.