Let \( f(x)\) be a real differentiable function such that \(f(0)=1\text{ and }\)\(f(\mathrm{x}+\mathrm{y})=f(\mathrm{x})…

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

[JEE Main 2025, 22 Jan (Shift 1)]