In fact, for two matrices <math>A, B \in \mathbb{C}^{n \times n}</math> we have <math>\log(AB) = \log(A) + \log(B)</math> if and only if <math>\operatorname{arg}(\mu_j) + \operatorname{arg}(\nu_j) \in (- \pi, \pi]</math>, where <math>\mu_j</math> is an [[eigenvalue]] of <math>A</math> and <math>\nu_j</math> is the corresponding [[eigenvalue]] of <math>B</math>.<ref>{{cite journal |last1=APRAHAMIAN |first1=MARY |last2=HIGHAM |first2=NICHOLAS J. |title=THE MATRIX UNWINDING FUNCTION, WITH AN APPLICATION TO COMPUTING THE MATRIX EXPONENTIAL |journal=SIAM J. MATRIX ANAL. APPL. |volume=35 |issue=1 |page=97 |doi=10.1137/130920137 |url=https://epubs.siam.org/doi/pdf/10.1137/130920137 |access-date=13 December 2022}}</ref> In particular, <math>\log(AB) = \log(A) + \log(B)</math> when ''A'' and ''B'' commute and are both [[Definite matrix|positive-definite]]. Setting ''B'' = ''A<sup>−1</sup>'' in this equation yields
*Substituting in this equation ''B'' = ''A<sup>−1</sup>'', one gets
:<math> \log{(A^{-1})} = -\log{(A)}.</math>
*Similarly, now for non-commuting ''A'' and ''B'',<ref>[https://www.ias.edu/sites/default/files/sns/files/1-matrixlog_tex(1).pdf Unpublished memo] by S Adler (IAS)</ref>
:<math>\log{(A+tB)} = \log{(A)}+t\int_0^\infty \!\! \! dz ~~\frac{I}{A+zI} B \frac{I}{A+zI} +O(t^2).</math>
