Fun(ctional) equation #2

Algebra Level 4

2f(x)=f(xy)+f(xy)2f(x)=f(xy)+f\left(\dfrac{x}{y}\right)

If the above equation is true for all x,yR+x,y\in \mathbb{R}^{+} and f(1)=0,f(1)=1f(1)=0, f'(1)=1, then find the closed form of f(1000) f(1000) to 3 decimal places.

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