10. (1991, 2nd "Hope" Math Competition) Let $\{x\}$ denote the smallest integer not less than the real number $x$. Then the value of $\left\{\log _{2} 1\right\}+\left\{\log _{2} 2\right\}+\left\{\log _{2} 3\right\}+\cdots+\left\{\log _{2} 1991\right\}$ is what?
10. By the definition of $\{x\}$
$$
\begin{array}{l}
\left\{\log _{2} 1\right\}=0, \\
\left\{\log _{2} 2\right\}=1, \\
\left\{\log _{2} 3\right\}=\left\{\log _{2} 4\right\}=2, \\
\left\{\log _{2} 5\right\}=\left\{\log _{2} 6\right\}=\left\{\log _{2} 7\right\}=\left\{\log _{2} 8\right\}=3, \\
\cdots \cdots \\
\left\{\log _{2} 513\right\}=\left\{\log _{2} 514\right\}=\cdots=\left\{\log _{2} 1024\right\}=10, \\
\left\{\log _{2} 1025\right\}=\cdots=\left\{\log _{2} 1991\right\}=11 .
\end{array}
$$

Then the required sum is
$$
0+1+2^{1} \cdot 2+2^{2} \cdot 3+2^{3} \cdot 4+\cdots+2^{9} \cdot 10+967 \cdot 11=19854 .
$$