${\log ^x} + {\log ^{\left( {x - 1} \right)}} = 1 + {\log ^2}$$ \Rightarrow {\log ^x} + {\log ^{\left( {x - 1} \right)}} - {\log ^2} = 1$${\log ^{\frac{{x\left( {x - 1} \right)}}{2}}} = 1 \Rightarrow \frac{{x\left( {x - 1} \right)}}{2} = 10$$ \Rightarrow \frac{{{x^2} - x}}{2} = 10 \Rightarrow {x^2} - x = 20$${x^2} - x - 20 = \left( {x - 5} \right)\left( {x + 4} \right)$$ = 0\begin{array}{*{20}{c}}{ \nearrow x - 5 = 0 \Rightarrow x = 5\,\,\,}\\{ \searrow x + 4 = 0 \Rightarrow x =  - 4}\end{array}$${x^2} + x = {5^2} + 5 = 25 + 5 = 30$