Unlike the scalar logarithm, there are no naturally-defined bases for the matrix logarithm; therefore, the matrix logarithm is always taken to be the natural logarithm. In general, there may be an infinite number of matrices $B$ satisfying $\exp(B)=A$ ; these are known as the logarithms of $A$ .

As for the scalar natural logarithm, the matrix logarithm can be defined as a power series when $A$ is a square matrix and , where is the Frobenius matrix norm. The logarithm this formula produces is known as the principal logarithm of $A$ . \begin{equation*} \log(A) = -\sum_{k=1}^\infty \frac{(I-A)^k}{k} =\log(I+X) = \sum_{k=1}^\infty \frac{(-1)^{k+1}}{k} X^k \end{equation*} Since this series expansion does not converge for all $A$ , it is not a global inverse function for the matrix exponential. In particular, $\exp^{\log A}=A$ only holds for , and $\log(\exp^A)=A$ only holds for .

There are other, more general methods of calculating the matrix logarithm. For example, see An Explicit Formula for the Matrix Logarithm.