#### Please Solve RD Sharma Class 12 Chapter 14 Inverse Trigonometric Function Exercise Fill in the Blanks Question 5 Maths Textbook Solution.

$\log_{2}e$

Hint:

You must know about mean value theorem.

Given:

$f\left ( x \right )=\log_{e}x,x\: \epsilon \left [ 1,2 \right ]$

Solution:

$f\left ( x \right )=\log_{e}x$

On differentiating we get,

$\Rightarrow {f}'\left ( x \right )=\frac{1}{x}$

$\Rightarrow {f}'\left ( c \right )=\frac{1}{c}$

Apply mean value theorem,

$\Rightarrow {f}'\left ( c \right )=\frac{\log_{e}2-\log_{e}1}{2-1}$

$\Rightarrow \frac{1}{c}=\log_{e}2-\log_{e}1$

$\Rightarrow \frac{1}{c}=\log_{e}2$                                                                                                                                  $\left [ \because \log_{e}1=0 \right ]$

$\Rightarrow \frac{1}{\log_{e}2}=c$

$\Rightarrow c=\log_{2}e$                                                                                                                    $\left ( \because \frac{1}{\log_{e}2}=\log_{2}e \right )$