7.179. $\log _{a^{2}} x^{2}+\log _{a}(x-1)=\log _{a} \log _{\sqrt{5}} 5$.
Solution.

Domain of definition: $\left\{\begin{array}{l}x>1, \\ 0<a \neq 1 .\end{array}\right.$

From the condition we have

$$
\log _{a} x+\log _{a}(x-1)=\log _{a} 2 \Rightarrow \log _{a} x(x-1)=\log _{a} 2
$$

from which $x^{2}-x-2=0 \Rightarrow x_{1}=2, x_{2}=-1 ; x_{2}=-1$ does not satisfy the domain of definition.

Answer: 2.