If $2 f(x y)=(f(x)) y+(f(y))^x \forall \quad x y \in \mathbf{R}$ and $f(1)=2$ then $\sum_{n=1}^9 f(n

Question

If $2 f(x y)=(f(x)) y+(f(y))^x \forall \quad x y \in \mathbf{R}$ and $f(1)=2$ then $\sum_{n=1}^9 f(n)=$

Professor PC Thomas Classes · Answer

$ \begin{aligned} & 	ext { If } y=1 	ext { then } 2 f(x)=f(x)+(f(1))^x \ & 	herefore f(x)=2^x \sum_{n=1}^9 f(n)=\sum_{n=1}^9 2^9=2^{10}-2 \end{aligned} $