How to expand this matrix formula?

Question

$L$ is a matrix, is it possible to expand this formula in terms of $\operatorname{Tr}L^n$? $$\log\operatorname{Tr}e^L$$

Answer 1

$$ \log Tr e^L = \log \left(Tr I_{n\times n} + Tr L + \frac{Tr (L^2)}{2!} + \dots\right)$$ $$= \log n + \log \left( 1 + \frac{Tr(L)}{n} + \frac{Tr(L^2)}{2! n} + \dots \right) $$

Using the expansion $\log(1 + x) = x - \frac{x^2}{2} + \frac{x^2}{3} - \dots$, you can obtain a series expression for what you want.