$14 \cdot 69$ Let $\{x\}$ denote the smallest integer not less than the real number $x$, then
$$
\left\{\log _{2} 1\right\}+\left\{\log _{2} 2\right\}+\left\{\log _{2} 3\right\}+\cdots+\left\{\log _{2} 1991\right\}
$$

What is the value of the above expression?
(2nd "Hope Cup" Mathematics Competition, 1991)
［Solution］By the definition of $\{x\}$
$$
\begin{array}{l}
\left\{\log _{2} 1\right\}=0, \\
\left\{\log _{2} 2\right\}=1, \\
\left\{\log _{2} 3\right\}=\left\{\log _{2} 4\right\}=2, \\
\left\{\log _{2} 5\right\}=\left\{\log _{2} 6\right\}=\left\{\log _{2} 7\right\}=\left\{\log _{2} 8\right\}=3,
\end{array}
$$
......
$$
\begin{array}{l}
\left\{\log _{2} 513\right\}=\left\{\log _{2} 514\right\}=\cdots=\left\{\log _{2} 1024\right\}=10, \\
\left\{\log _{2} 1025\right\}=\cdots=\left\{\log _{2} 1991\right\}=11 .
\end{array}
$$

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