We obtain analytical expressions for the eigenvalues and eigenstates of a family of exponential-type potential wells. The supersymmetry formalism applied to quantum mechanics is summarized and illustrated by producing from this family of potentials another class of exact solutions made of their isospectral partners. A subset of the supersymmetric partners provides a class of exactly solvable double well potentials. The exact solutions of these potentials are used to test the robustness and accuracy of different approximation methods. We determine the ground state through a variational method applied to a set of trial functions and the entire spectrum using the WKB, JWKB, and its supersymmetric extension formulas. We comment on the importance of the Maslov index and on the range of validity of these semiclassical quantization approaches.