CEE Kerala Engineering CEE Kerala Engineering Solved Paper-2003

  • question_answer If\[f(x)={{\log }_{x}}({{\log }_{e}}x),\]then\[f(x)\]at\[x=e\]is equal to:

    A)  1                                            

    B)  2                            

    C)  0                            

    D)         \[e\]

    E)  \[1/e\]

    Correct Answer: E

    Solution :

    \[\because \] \[f(x)={{\log }_{x}}({{\log }_{e}}x)=\frac{{{\log }_{e}}{{\log }_{e}}x}{{{\log }_{e}}x}\] \[\therefore \] \[f(x)=\frac{\left[ \begin{align}   & {{\log }_{e}}x\frac{1}{{{\log }_{e}}x}.\frac{1}{x}- \\  & \,\,\,\,\,\,\,\,\,\,\,\,\log {{\log }_{e}}x.\frac{1}{x} \\ \end{align} \right]}{{{({{\log }_{e}}x)}^{2}}}\] \[\Rightarrow \]               \[f(x)=\frac{1-{{\log }_{e}}{{\log }_{e}}x}{x{{({{\log }_{e}}x)}^{2}}}\] \[\Rightarrow \]               \[f(e)=\frac{1-{{\log }_{e}}{{\log }_{e}}e}{e({{\log }_{e}}e)}=\frac{1-{{\log }_{e}}1}{e}\]                                 \[=\frac{1}{e}\]

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