${\left( {{{\log }^x}} \right)^2} + \log x + 1 = \frac{7}{{{{\log }^{\frac{x}{{10}}}}}}$$ \Rightarrow 2{\log ^x} + {\log ^x} + 1 = $$\frac{7}{{{{\log }^x} - {{\log }^{10}}}} \to 3{\log ^x} + 1 = \frac{7}{{{{\log }^x} - 1}}$$ \Rightarrow \left( {3{{\log }^x} + 1} \right)\left( {{{\log }^x} - 1} \right) = 7$$ \Rightarrow 3{\log ^x} \times {\log ^x} - 3{\log ^x} + {\log ^x} - 1 = 7$$ \Rightarrow 3{\log ^{{x^2}}} - 2{\log ^x} - 8 = 0$${\log ^x} = A \Rightarrow 3{A^2} - 2A - 8 = 0$$ \Rightarrow \left( {3A - 6} \right)\left( {3A + 4} \right) = 0$$\begin{array}{*{20}{c}}{ \nearrow 3A - 6 = 0 \Rightarrow 3A = 6 \Rightarrow A = 2\,\,\,\,\,\,\,\,}\\{ \searrow 3A + 4 = 0 \Rightarrow 3A =  - 4 \Rightarrow A =  - \frac{4}{3}}\end{array}$