1. Given the function $f(x)=\log _{a}\left(a x^{2}-x+\frac{1}{2}\right)$ is always positive on the interval $[1,2]$. Then the range of the real number $a$ is
1. $\frac{1}{2}\frac{3}{2}$.
From $a x^{2}-x+\frac{1}{2}>0$, we get
$$
a>\frac{x-\frac{1}{2}}{x^{2}}=h(x) \text {. }
$$

Since $h^{\prime}(x)=\frac{1-x}{x^{3}}$, when $1 \leqslant x \leqslant 2$, $h^{\prime}(x)\frac{1}{2}$, and $a \neq 1$.
When $\frac{1}{2}0$, we know $\frac{1}{2}1$, $f(x)$ is an increasing function on $[1,2]$.
From $f(1)>0$, we know $a>\frac{3}{2}$.