Let $\mu_i,\nu_i\in\mathbb{R}^{n\times n}$ for $i\in{\{1,\dots,I\}}$, and let $\sigma\in\mathbb{R}$. Assume that $\log{\prod_{i=1}^I{\exp{(\mu_i)}}}$ exists with real entries.
It must be the case that for some matrix $M\in\mathbb{R}^{n\times n}$:
$$\log{\prod_{i=1}^I{\exp{(\mu_i+\sigma\nu_i)}}}=\log{\prod_{i=1}^I{\exp{(\mu_i)}}}+M\sigma+O(\sigma ^2 )$$
as $\sigma\rightarrow 0$.
What is $M$?
The derivative of an exponential of an arbitrary matrix function seems to be rather messy, but I was hoping there would be a simpler answer thanks to linearity, and only being interested in first order terms.
Further, by vectorising and using the properties of Kronecker products and commutation matrices, can one express $M$ in the form:
$$\operatorname{vec}{M}=\sum_{i=1}^I{f_i(\mu_1,\dots,\mu_I)\operatorname{vec}{\nu_i}}\space\space\space\space\space ?$$
P.S. I've tagged this with the Lie group and Lie algebra tags as it's abundantly clear that much of the relevant results are in the more general case. But my knowledge of Lie theory is minimal, so it would be appreciated if answers could keep things as specific to real matrices as possible.