7.247. $\sqrt{\log _{5} x}+\sqrt[3]{\log _{5} x}=2$.
Solution.

Domain of definition: $\log _{5} x \geq 0$ or $x \geq 1$.

Rewrite the equation as $\sqrt[6]{\left(\log _{5} x\right)^{3}}+\sqrt[6]{\left(\log _{5} x\right)^{2}}-2=0$. Let $\sqrt[6]{\log _{5} x}=y$. The equation in terms of $y$ becomes

$$
y^{3}+y^{2}-2=0 \Leftrightarrow\left(y^{3}-1\right)+\left(y^{2}-1\right)=0 \Leftrightarrow
$$

$\Leftrightarrow(y-1)\left(y^{2}+y+1\right)+(y-1)(y+1)=0 \Leftrightarrow(y-1)\left(y^{2}+2 y+2\right)=0$, from which $y-1=0$, since $y^{2}+2 y+2>0$. We get $\sqrt[6]{\log _{5} x}=1, \log _{5} x=1$, $x=5$.

Answer: 5.