This work studies an optimization approach for designing tone reproduction curve (TRC) based tone mapping operators for the display of high dynamic range (HDR) images in low dynamic range (LDR) reproduction media. Previous work has shown that the tone mapping problem can formulated as that of optimizing a two-term cost function where adjusting the relative weightings of the two terms allows users to interactively control the appearance of the output image. However, only heuristic solutions to the tone mapping objective function have been found in past research. The main contribution of this paper is the re-formulation of the objective function to allow the introduction of a closed-form solution to the two-term tone mapping objective function. The new solution has simplified previous heuristic solutions and made this approach mathematically more elegant, computationally faster and practically easier to implement.