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In this work, new closed-form formulas for the matrix exponential areprovided. Our method is direct and elementary, it gives tractable andmanageable formulas not current in the extensive literature on this essentialsubject. Moreover, others are recuperated and generalized. As a consequence, weeasily obtain the Chevalley{Jordan decomposition and the spectral projectionsof any matrix. In addition, closed-form expressions for the arbitrary positivepowers of matrices and their Drazin inverses are presented. Using theseresults, an elegant explicit formula for logarithm of matrices is obtained.Several particular cases and examples are formulated to illustrate the methodspresented in this paper.

Some Explicit Formulas for Matrix Exponential, Matrix Logarithm, the $n$th Power of Matrices and their Drazin Inverses