Question

Let $e^{f\left(x\right)}=\log x$, if $g\left(x\right)$ is the inverse function of $f\left(x\right)$ then $g’\left(x\right)=$

• A. $e^x$
• B. $e^{x+e^x}$
• C. $\log x$
• D. $x$

Answer 1

The correct option is B

$e^{x+e^x}$

Explanation for correct option:

The given equation is:

$e^{f\left(x\right)}=\log x......\left(1\right)$

$f^{-1}\left(x\right)=g\left(x\right)$

Take $\log$ on both sides of the equation $\left(1\right)$

$$\begin{gathered} \Rightarrow \log e^{f\left(x\right)}=\log \left(\log x\right) \\ \Rightarrow f\left(x\right)=\log \left(\log x\right) \end{gathered}$$

Let,

$$\begin{gathered} f\left(x\right)=y \\ \Rightarrow x=f^{-1}\left(y\right) \end{gathered}$$

Then substitute $f\left(x\right)$.

$$\begin{gathered} \log \left(logx\right)=y \\ \Rightarrow \log x=e^y \\ \Rightarrow x=e^{e^y} \end{gathered}$$

Since

$$\begin{gathered} f^{-1}\left(y\right)=e^{e^y} \\ \Rightarrow f^{-1}\left(x\right)=e^{e^x} \end{gathered}$$

From the given data $f^{-1}\left(x\right)=g\left(x\right)$ so that

$$\begin{gathered} g\left(x\right)=e^{e^x} \\ \Rightarrow g\text{'}\left(x\right)=e^{e^x}.e^x \\ \Rightarrow g\text{'}\left(x\right)=e^{e^x+x} \end{gathered}$$

Hence, the correct answer is option (B).