Question

If $3f\left(x\right)-f\left(\frac{1}{x}\right)=\log x^4$, then $f\left(e^{-x}\right)$ is

• A. $1+x$
• B. $\frac{1}{x}$
• C. $x$
• D. $-x$

Answer 1

The correct option is D

$-x$

Explanation for the correct option:

Step 1. Find the value of $f\left(e^{-x}\right)$:

Given, $3f\left(x\right)-f\left(\frac{1}{x}\right)=\log x^4$

$\Rightarrow$ $3f\left(x\right)–f\left(\frac{1}{x}\right)=4\log x$ …..(i) $\left[\mathbf{\because }\mathit{l}\mathit{o}\mathit{g}\mathbf{}{\mathit{m}}^{\mathbf{x}}\mathbf{}\mathbf{=}\mathbf{}\mathit{x}\mathbf{}\mathit{l}\mathit{o}\mathit{g}\mathbf{}\mathit{m}\right]$

Step 2. Replace $x$ by $\frac{1}{x}$:

$\Rightarrow$$3f\left(\frac{1}{x}\right)–f\left(x\right)=4\log \left(\frac{1}{x}\right)$

$\Rightarrow$$3f\left(\frac{1}{x}\right)–f\left(x\right)=-4\log x$ …..(ii) $\left[\mathbf{\because }\mathit{l}\mathit{o}\mathit{g}\mathbf{}\mathit{a}\mathbf{}\mathbf{-}\mathbf{}\mathit{l}\mathit{o}\mathit{g}\mathbf{}\mathit{b}\mathbf{}\mathbf{=}\mathbf{}\mathit{l}\mathit{o}\mathit{g}\mathbf{}\left(\frac{a}{b}\right)\right]$

Multiply equation(i) by 3:

$\Rightarrow$$9f\left(x\right)–3f\left(\frac{1}{x}\right)=12\log x$ …..(iii)

Step 3. Add equation(ii) and (iii):

$\Rightarrow$ $8f\left(x\right)=8\log x$

$\Rightarrow$ $f\left(x\right)=\log x$

$\Rightarrow$$\begin{array}{rcl}f\left(e^{-x}\right)& =& \log e^{-x}\\ & =& -x\end{array}$

$\therefore$$f\left(e^{-x}\right)$ is equal to $\mathbf{-}\mathit{x}$.

Hence, Option ‘D’ is Correct.