Existence of Matrix Logarithm

Theorem
Let $T$ be an $n \times n$ matrix. Then there is a real matrix $S$ such that $e^S=T$ $T$ is not a singular matrix and, for every negative eigenvalue $\lambda$ of $T$ and for every positive integer $k$, the Jordan form of $T$ has an even number of $k \times k$ blocks associated with $\lambda$.