jmc

For $1\le N\le 1024,$ the possible values of $\lfloor {\log }_{2}N\rfloor$ are 0, 1, 2, $\dots ,$ 10. For a given value of $k,$ $0\le k\le 10,$ $$\lfloor {\log }_{2}N\rfloor =k$$for $N=2^k,$ $2^{k+1},$ $\dots ,$ $2^{k+1}-1,$ for $2^k$ possible values. The only exception is $k=10$: $\lfloor {\log }_{2}N\rfloor =10$ only for $N=1024.$

Hence, the sum we seek is $$S=1\cdot 0+2\cdot 1+2^2\cdot 2+2^3\cdot 3+\cdots +2^8\cdot 8+2^9\cdot 9+10.$$Then $$2S=2\cdot 0+2^2\cdot 1+2^3\cdot 2+2^4\cdot 3+\cdots +2^9\cdot 8+2^{10}\cdot 9+20.$$Subtracting these equations, we get $$\begin{array}{rl}S& =10+2^{10}\cdot 9-2^9-2^8-\cdots -2^2-2\\ & =10+2^{10}\cdot 9-2(2^8+2^7+\cdots +2+1)\\ & =10+2^{10}\cdot 9-2(2^9-1)\\ & =\boxed{8204}.\end{array}$$