Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function with $f(0) = 1$ and satisfying the equation $f(x+y) = f(x)f'(y) + f'(x)f(y)$ for all $x, y \in \mathbb{R}$ Then, the value of $\log_e (f(4))$ is ________.

$f(x+y) = f(x)f'(y) + f'(x)f(y), \quad f(0)=1$

$\text{To find: } \log_e(f(4))$

$y=0:\ f(x) = f(x)f'(0) + f'(x)\cdot f(0) = f(x)f'(0) + f'(x)$

$f'(x) = f(x)\,(1-f'(0))$

$\text{let } c = 1-f'(0):\quad f'(x) = c\,f(x)$

$\int \dfrac{df}{f} = \int c\,dx \;\Rightarrow\; \ln f = cx + K$

$f(x) = e^{cx} \quad (\text{using } f(0)=1)$

$e^{c(x+y)} = e^{cx}(c\,e^{cy}) + (c\,e^{cx})e^{cy} = 2c\,e^{c(x+y)}$

$1 = 2c \;\Rightarrow\; c = \dfrac{1}{2}$

$f(x) = e^{x/2}$

$f(4) = e^{4/2} = e^{2}$

$\log_e(f(4)) = \log_e(e^{2}) = 2$

$\boxed{\log_e(f(4)) = 2}$