So I've worked through a couple examples which were straight forward.
Matrices like $A = \left[\begin{array}{cc} 4 & 1\\ 0 & e\end{array}\right]$, were easy because $A$ is diagonalizable.
The only non-diagonalizable example we covered in class were of the form
$\lambda I + N$, where $N^r = 0$ for some positive integer $r$, then we used the formula
$\log(\lambda I + N) = \log(\lambda I) + \sum\limits_{n=1}^{r-1}\frac{(-N)^{n}}{j\lambda^{j}}$.
How can you calculate the log if it doesn't fall under one of these two forms?