A value of $c$ for which conclusion of Mean Value Theorem holds for the function $f(x)=\log _{e} x$

Question

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

Professor PC Thomas Classes · Answer

(c) : By LMVT,\ $f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}=\frac{f(3)-f(1)}{3-1}$\ $f^{\prime}(c)=\frac{\log _{e} 3-\log _{e} 1}{2}=\frac{1}{2} \log _{e} 3$\ $\Rightarrow \quad \frac{1}{c}=\frac{1}{2} \log _{e} 3=\frac{1}{2 \log _{3} e}$\ $	herefore c=2 \log _{3} e$.\