Let $f : R \to R$ 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 $x, y ϵ R$ .
Then, value of ${log}_{e} (f (4))$ is ________.

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Solution

$P (x, y) : f (x + y) = f (x) f^{'} (y) + f^{'} (x) f (y) \forall x, y ϵ R$

$P (0, 0) : f (0) = f (0) f^{'} (0) + f^{'} (0) f (0)$

$\Rightarrow 1 = 2 f^{'} (0)$

$\Rightarrow f^{'} (0) = \frac{1}{2}$

$P (x, 0) : f (x) = f (x) . f^{'} (0) + f^{'} (x) . f (0)$

$\Rightarrow f (x) = \frac{1}{2} f (x) + f^{'} (x)$

$\Rightarrow f^{'} (x) = \frac{1}{2} f (x)$

$\Rightarrow f (x) = e^{\frac{1}{2} x}$

$\Rightarrow l n (f (4)) = 2$ .

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