A) \[lo{{g}_{3}}e\]
B) \[lo{{g}_{e}}3\]
C) \[2lo{{g}_{3}}e\]
D) \[\frac{1}{2}{{\log }_{3}}e\]
Correct Answer: C
Solution :
[c] Using Lagrange?s Mean Value Theorem Let f(x) be a function defined on [a, b] then, \[f'(c)=\frac{f(b)-f(a)}{b-a}...(i)\] \[c\in [a,b]\therefore Given\,f(x)={{\log }_{e}}x\therefore f'(x)=\frac{1}{x}\] \[\therefore \] equation (i) become \[\frac{1}{c}=\frac{f(3)-f(1)}{3-1}\] \[\Rightarrow \frac{1}{c}=\frac{{{\log }_{e}}3-{{\log }_{e}}1}{2}=\frac{{{\log }_{e}}3}{2}\] \[\Rightarrow c=\frac{2}{{{\log }_{e}}3}\Rightarrow c=2\,\,{{\log }_{3}}e\]
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