This problem requires us to find the domain of a function which is a sum of a logarithmic function and an inverse cosine function. The domain of such a composite function is the intersection of the domains of its individual components.

Key Concepts for Domain Calculation:

1. Domain of $\log_e(u)$: For $\log_e(u)$ to be defined, its argument $u$ must be strictly positive. That is, $u > 0$.
2. Domain of $\cos^{-1}(v)$: For $\cos^{-1}(v)$ to be defined, its argument $v$ must be in the interval $[-1, 1]$. That is, $-1 \leq v \leq 1$.

Let the given function be $f(x