1. Given the sets $A=\left\{x \mid \log _{2}(x-1)<1\right\}, B=\{x|| x-a \mid<2\}$, if $A \cap B \neq \varnothing$, then the range of real number $a$ is $\qquad$ .
$A=(1,3), B=(a-2, a+2)$, so $a-2<1<a+2$ or $a-2<3<a+2$ $\Rightarrow-1<a<3$ or $1<a<5 \Rightarrow-1<a<5$.

