$\operatorname{tr}((1 -\varepsilon)^A)= \operatorname{tr}(A)$?

Question

Let $\varepsilon \geq 0 $ for matrices $A \in\mathbb R^{n \times n}$

$$\operatorname{tr}((1 -\varepsilon)^A)= \operatorname{tr}(A) \text{ ?}$$

where $(1 -\varepsilon)^A = e^{\log(1-\varepsilon)\cdot A}$ how to see that ?