Let $f:(0, \infty) \rightarrow \mathbb{R}$ be the continuous function such that $f(x)=2+\frac{g(x)}{x}$ for all $x>0$, where $g(x)=\int_{1}^{x} f(t) d t$ for all $x>0$. Then $f(2)$ is equal to
- $2+\log _{e} 2$
- $2-\log _{e} 2$
- $2+\log _{e} 4$
- $2-\log _{e} 4$
