7.053. $\lg (\lg x)+\lg \left(\lg x^{3}-2\right)=0$.

Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.

7.053. $\lg (\lg x)+\lg \left(\lg x^{3}-2\right)=0$.
Solution.

Domain of definition: $\left\{\begin{array}{l}\lg x>0, \\ \lg x^{3}-2>0,\end{array} \quad x>\sqrt[3]{100}\right.$.

From the condition we have

$\lg \left(\lg x \cdot\left(\lg x^{3}-2\right)\right)=0, \quad \lg x(3 \lg x-2)=1$,

$3 \lg ^{2} x-2 \lg x-1=0$.

Solving this equation as a quadratic equation in terms of $\lg x$, we find $(\lg x)_{1}=-\frac{1}{3}$, hence $x_{1}=\frac{1}{\sqrt[3]{10}}$, or $(\lg x)_{2}=1$, hence $x_{2}=10 ; x_{1}=\frac{1}{\sqrt[3]{10}}$ does not satisfy the domain of definition.

Answer: 10.