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Note that the exponential map is a local diffeomorphism between a neighborhood ''U'' of the zero matrix <math> \underline{0} \in \mathfrak{g}</math> and a neighborhood ''V'' of the identity matrix <math>\underline{1}\in G</math>.<ref>{{harvnb|Hall|2015}} Theorem 3.42</ref>
Thus the (matrix) logarithm is well-defined as a map,
:<math> \log:
An important corollary of [[Jacobi's formula]] then is
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