How does the following expression scale with respect to $\epsilon$?

Question

Let $\mathbf{V}$ be a $2N \times 2N$ positive definite matrix.

Let $\mathbf{V}^\star=\mathbf{V}+E$ where $||E||_\infty\leq\epsilon$ and $|\epsilon|<<1$. How does the following expression scale with respect to $\epsilon$?

$\log\left(\max\left(||\mathbf{V}^\star||_\infty,\left(||\mathbf{V}^{\star^{-1}}||^{-1}_\infty-1\right)/2\right)\right)$