2. Given the function $f(x)$ $=\log _{a}\left(\frac{1}{2} a x^{2}-x+\frac{1}{2}\right)$ is always positive on the interval $[2,3]$, then the range of the real number $a$ is $\qquad$ .
The graph of the function $g(x)=\frac{1}{2} a x^{2}-x+\frac{1}{2}$ is symmetric about the line $x=\frac{1}{a}$. When $a>1$, $g(x)$ is monotonically increasing on $[2,3]$, so $\frac{1}{2} a \times 4-2+\frac{1}{2}>1 \Rightarrow a>\frac{5}{4}$;
When $00\end{array}\right.$ $\Rightarrow \frac{5}{9}0, \\ \frac{1}{2} a \times 9-3+\frac{1}{2}0$, thus there is no solution in this case. In summary, the range of real number $a$ is $\left(\frac{3}{4}, \frac{7}{9}\right) \cup\left(\frac{5}{4},+\infty\right)$.