A) \[x=3\]
B) \[x=4\sqrt{3}\]
C) \[x=9\]
D) \[x=\sqrt{3}\]
Correct Answer: D
Solution :
\[{{\log }_{\sqrt{3}}}x+{{\log }_{\sqrt[4]{3}}}x+{{\log }_{\sqrt[6]{3}}}x+......+{{\log }_{\sqrt[16]{3}}}x=36\] \[\Rightarrow \]\[\frac{1}{{{\log }_{x}}\sqrt{3}\,}+\frac{1}{{{\log }_{x}}\sqrt[4]{3}}+\frac{1}{{{\log }_{x}}\sqrt[6]{3}}+...+\frac{1}{{{\log }_{x}}\sqrt[16]{3}}=36\] \[\Rightarrow \] \[\frac{1}{(1/2){{\log }_{x}}3}+\frac{1}{(1/4){{\log }_{x}}3}+\frac{1}{(1/6){{\log }_{x}}3}+............\] \[............+\frac{1}{(1/16){{\log }_{x}}3}=36\] \[\Rightarrow \] \[({{\log }_{3}}x)(2+4+6+.....+16)=36\] \[\Rightarrow \] \[({{\log }_{3}}x)\frac{8}{2}[2+16]=36\]\[\Rightarrow \]\[{{\log }_{3}}x=\frac{1}{2}\] \[\Rightarrow \]\[x={{3}^{1/2}}\]\[\Rightarrow x=\sqrt{3}\].
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