Let \(f: \mathbb{R} \rightarrow[0, \infty)\) be a continuous function such that \[f(x+y)=f(x) f(y)\] for all \(x, y \in \mathbb{R}\). Suppose that \(f\) is differentiable at \(x=1\) and \[\left.\frac{d f(x)}{d x}\right|_{x=1}=2\] Then, the value of \(f(1) \log _{e} f(1)\) is