Question

If $x>0$ and ${\log }_{3}x+{\log }_3\sqrt{x}+{\log }_3\sqrt[4]{x}+.......=4$, then $x$ equals

• A. $9$
• B. $81$
• C. $1$
• D. $27$

Answer 1

The correct option is A

$9$

Explanation for the correct answer:

Finding the value of $x$:

Given,

$$\begin{gathered} {\log }_3\mathrm{x}+{\log }_3\sqrt{\mathrm{x}}+{\log }_3\sqrt[4]{\mathrm{x}}+.......=4 \\ {\log }_3\mathrm{x}+\frac{1}{2}{\log }_3\mathrm{x}+\frac{1}{4}{\log }_3\mathrm{x}+......=4\mathbf{}\mathbf{[}\mathbf{\because }\mathbf{}\mathbf{log}\mathbf{}{\mathbf{x}}^{\mathbf{n}}\mathbf{=}\mathbf{n}\mathbf{}\mathbf{log}\mathbf{}\mathbf{x}\mathbf{]} \\ \Rightarrow {\log }_3\mathrm{x}\left(1+\frac{1}{2}+\frac{1}{4}+....\right)=4 \\ \Rightarrow {\log }_3\mathrm{x}\left(\frac{1}{1-{\displaystyle \frac{1}{2}}}\right)=4\mathbf{[}\mathbf{\because }\mathbf{}{\mathbf{S}}_{\mathbf{n}}\mathbf{=}\frac{\mathbf{}\mathbf{a}}{\mathbf{1}\mathbf{-}\mathbf{r}}\mathbf{,}\mathbf{where}\mathbf{}\mathbf{a}\mathbf{}\mathbf{is}\mathbf{}\mathbf{first}\mathbf{}\mathbf{term}\mathbf{}\mathbf{and}\mathbf{}\mathbf{r}\mathbf{}\mathbf{is}\mathbf{}\mathbf{common}\mathbf{}\mathbf{ratio}\mathbf{.}\mathbf{]} \\ \Rightarrow {\log }_3\mathrm{x}\left(2\right)=4 \\ \Rightarrow {\log }_3\mathrm{x}=\frac{4}{2} \\ \Rightarrow {\log }_3\mathrm{x}=2 \\ \Rightarrow \mathrm{x}=3^2\mathbf{[}\mathbf{\because }\mathbf{if}\mathbf{}{\mathbf{log}}_{\mathbf{n}}\mathbf{x}\mathbf{=}\mathbf{k}\mathbf{,}\mathbf{}\mathbf{then}\mathbf{}\mathbf{x}\mathbf{=}{\mathbf{k}}^{\mathbf{n}}\mathbf{]} \\ \Rightarrow \mathrm{x}=9 \end{gathered}$$

Hence, the correct option is A.