Question

If $\frac{1}{{\log }_3\mathrm{\pi }}+\frac{1}{{\log }_4\mathrm{\pi }}>x$, then $x$ will be

• A. $3$
• B. $2$
• C. $4$
• D. None of these

Answer 1

The correct option is B

$2$

STEP 1: Using the property of log function that $\mathit{l}\mathit{o}{\mathit{g}}_{\mathbf{b}}\mathit{a}\mathbf{=}\frac{\mathbf{l}\mathbf{o}\mathbf{g}\mathbf{a}}{\mathbf{l}\mathbf{o}\mathbf{g}\mathbf{b}}$.

$$\begin{gathered} \Rightarrow {\log }_3\mathrm{\pi }=\frac{\mathrm{log\pi }}{\log 3} \\ \Rightarrow {\log }_4\mathrm{\pi }=\frac{\mathrm{log\pi }}{\log 4} \end{gathered}$$

STEP 2: Substituting values of $\mathit{l}\mathit{o}{\mathit{g}}_{\mathbf{3}}\mathbf{\pi }\mathbf{,}\mathbf{}{\mathbf{log}}_{\mathbf{4}}\mathbf{\pi }$ obtained in step 1 in $\frac{\mathbf{1}}{\mathbf{l}\mathbf{o}{\mathbf{g}}_{\mathbf{3}}\mathbf{\pi }}\mathbf{+}\frac{\mathbf{1}}{\mathbf{l}\mathbf{o}{\mathbf{g}}_{\mathbf{4}}\mathbf{\pi }}$ and simplify.

$\begin{array}{rcl}& \Rightarrow & \frac{1}{{\log }_3\mathrm{\pi }}+\frac{1}{{\log }_4\mathrm{\pi }}=\frac{\log 3}{\mathrm{log\pi }}+\frac{\log 4}{\mathrm{log\pi }}\\ & =& \frac{\log 3+\log 4}{\mathrm{log\pi }}\\ & =& \frac{\log 12}{\mathrm{log\pi }}\\ & =& {\log }_{\mathrm{\pi }}12\end{array}$

STEP 3: Simplify $\frac{\mathbf{1}}{\mathbf{l}\mathbf{o}{\mathbf{g}}_{\mathbf{3}}\mathbf{\pi }}\mathbf{+}\frac{\mathbf{1}}{\mathbf{l}\mathbf{o}{\mathbf{g}}_{\mathbf{4}}\mathbf{\pi }}\mathbf{>}\mathit{x}$ using the value obtained in step 2.

$$\begin{gathered} \Rightarrow \frac{1}{{\log }_3\mathrm{\pi }}+\frac{1}{{\log }_4\mathrm{\pi }}>x \\ \Rightarrow {\log }_{\mathrm{\pi }}12>x \\ \Rightarrow 12>{\mathrm{\pi }}^{\mathrm{x}} \end{gathered}$$

Since $$\begin{gathered} 12>{\mathrm{\pi }}^2 \\ \Rightarrow \mathrm{x}=2 \end{gathered}$$.

Hence, option (B) is correct.