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Abstract. The most popular method for computing the matrix logarithm is the inverse scaling and squaring method, which is the basis of the recent algorithm of Al-Mohy and Higham [SIAM J. Sci. Comput., 34 (2012), pp. C152–C169]. For real matrices we develop a version of the latter algo-rithm that works entirely in real arithmetic and is twice as fast as and more accurate than the original algorithm. We show that by differentiating the algorithms we obtain backward stable algorithms for computing the Fréchet derivative. We demonstrate experimentally that our two algorithms are more accurate and efficient than existing algorithms for computing the Fréchet derivative and we also show how the algorithms can be used to produce reliable estimates of the condition number of the matrix logarithm