Let f(x) be a real differentiable function such that f(0)=1 and f(x+y)=f(x) f^′(y)+f^′(x) f(y) for all x, y ∈ R. Then Σ_n =1^100 log_e f( n ) is…

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