Let $f(x)$ be a real differentiable function such that $f(0)=1$ and
$f(x+y)=f(x){{f}^{\prime }}(y)+{{f}^{\prime }}(x)f(y)$ for all $x,y\in \mathbf{R}$. Then $\sum\limits_{{\text{n}=1}}^{{100}}{{{{{\log }}_{\text{e}}}}}f(\text{n})$ is equal to
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