• # question_answer The value of $x$ satisfying${{\log }_{a}}x+{{\log }_{\sqrt{a}}}x+{{\log }_{3\sqrt{a}}}x+.........{{\log }_{a\sqrt{a}}}x=\frac{a+1}{2}$ will be A) $x=a$ B) $x={{a}^{a}}$ C) $x={{a}^{-1/a}}$ D) $x={{a}^{1/a}}$

${{\log }_{a}}x+2{{\log }_{a}}x+.......+a{{\log }_{a}}x=\frac{a+1}{2}$ $\Rightarrow$ ${{\log }_{a}}x(1+2+........+a)=\frac{a+1}{2}$ $\Rightarrow$  ${{\log }_{a}}x.\frac{a(a+1)}{2}=\frac{a+1}{2}$$\Rightarrow$$({{10}^{12}}+{{10}^{11}}+......+1)$.