7.250. $\log _{a} x+\log _{\sqrt{a}} x+\log _{\sqrt[3]{a^{2}}} x=27$.
Solution.

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

Switch to base $a$. We have $\log _{a} x+2 \log _{a} x+\frac{3}{2} \log _{a} x=27 \Leftrightarrow$ $\log _{a} x=6$, from which $x=a^{6}$.

Answer: $a^{6}$, where $0<a \neq 1$.