The current algorithms use either the full form or the Schur decomposition of the matrix in the inverse scaling and squaring method to compute the matrix logarithm. The inverse scaling and squaring method consists of two main calculations: taking a square root and evaluating the Padé approximants. In this work, we suggest using the structure preserving iteration as an alternative to Denman-Beavers iteration for taking a square root. Numerical experiments show that while using the structure preserving square root iteration in the inverse scaling and squaring method preserves the Hamiltonian structure of matrix logarithm, Denman-Beavers iteration and Schur decomposition cause a structure loss.