A) 9
B) 81
C) 1
D) 27
Correct Answer: B
Solution :
\[{{\log }_{3}}x+{{\log }_{3}}\sqrt{x}+{{\log }_{3}}\sqrt[4]{x}+{{\log }_{3}}\sqrt[8]{x}+.....=4\] \[\Rightarrow \]\[{{\log }_{3}}x+\frac{1}{2}{{\log }_{3}}x+\frac{1}{4}{{\log }_{3}}x+\frac{1}{8}{{\log }_{3}}x\] \[+....=4\] \[\Rightarrow \]\[{{\log }_{3}}x\left[ 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+.... \right]=4\] \[\Rightarrow \] \[{{\log }_{3}}x\left[ \frac{1}{1-1/2} \right]=8\] \[\Rightarrow \] \[{{\log }_{3}}x=4\] \[\Rightarrow \] \[x={{3}^{4}}=81\]
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