9. (16 points) Let $P\left(x+a, y_{1}\right), Q\left(x, y_{2}\right), R\left(2+a, y_{3}\right)$ be three distinct points on the graph of the function $f(x)=\log _{2}(x-a)$, and satisfy $y_{1}+y_{3}=2 y_{2}$. The real number $x$ that satisfies this condition is unique. Find the range of the real number $a$.


9. It is known that $y_{1}=\log _{2} x, y_{2}=\log _{2}(x-a), y_{3}=1$. From $y_{1}+y_{3}=2 y_{2}$, we get $1+\log _{2} x=$ $2 \log _{2}(x-a)$. This equation is equivalent to
$$
\left\{\begin{array}{l}
x>a, \\
2 x=(x-a)^{2} .
\end{array}\right.
$$
(1) When $\Delta=0$, i.e., $a=-\frac{1}{2}$, the equation has a unique real root $x=\frac{1}{2}$.
(2) When $\Delta>0$, i.e., $a>-\frac{1}{2}$, the equation has two real roots $x=a+1 \pm \sqrt{2 a+1}$.

Clearly, $x=a+1+\sqrt{2 a+1}>a$ satisfies the condition, so we should have $x=a+1-\sqrt{2 a+1} \leqslant a$, solving this gives $a \geqslant 0$. In summary, the range of real numbers $a$ is $\left\{-\frac{1}{2}\right\} \cup(0,+\infty)$.