1. If $\log _{a} b \in \mathbf{Z}$ and $\log _{a} \frac{1}{b}>\log _{a} \sqrt{b}>\log _{b^{2}} a^{2}$, then among the following 4 conclusions:
(1) $\frac{1}{b}>\sqrt{b}>a^{2}$
(2) $\log _{a} b+\log _{a} a=0$
(3) $0<a<b<1$
(4) $a b=1$
The number of correct conclusions is ( ).
A. 1
B. 2
C. 3
D. 4
1. A

Let $\log _{a} b=k \in Z$. Then $-k>\frac{1}{2} k>\frac{2}{k}$, solving this yields $-2<k<0$ so $k=-1$.