Question

A value of c for which conclusion of Mean Value Theorem holds for the function $f\left(x\right)={\log }_{e}x$ on the interval $[1,3]$ is

• A. ${\log }_{3}e$
• B. ${\log }_{e}3$
• C. $2{\log }_{3}e$
• D. $\frac{1}{2}{\log }_{e}3$

Answer 1

Answer: (C)

$2{\log }_{3}e$

By L.M.V.T., $f^{\prime }\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}=\frac{f\left(3\right)-f\left(1\right)}{3-1}$

$f^{\prime }\left(c\right)=\frac{{\log }_{e}3-{\log }_{e}1}{2}=\frac{1}{2}{\log }_{e}3$

$\Rightarrow$ $\frac{1}{c}=\frac{1}{2}{\log }_{e}3=\frac{1}{2{\log }_{3}e}$

$\therefore$ $c=2{\log }_{3}e$