A) \[e\]
B) \[-e\]
C) \[{{e}^{2}}\]
D) \[{{e}^{-1}}\]
Correct Answer: D
Solution :
\[\because \] \[f(x)={{\log }_{x}}(\log x)=\frac{\log (\log x)}{\log x}\] \[\therefore \] \[f(x)=\frac{\log x.\frac{1}{\log x}.\frac{1}{x}-\log (\log x).\frac{1}{x}}{{{(\log x)}^{2}}}\] \[=\frac{1-\log (\log x)}{x{{(\log x)}^{2}}}\] Now, \[f(e)=\frac{1-\log (\log e)}{e{{(\log e)}^{2}}}=\frac{1-\log (1)}{e}=\frac{1}{e}\]
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