Question

If $f(x) = \log_5\log_3x$, then $f'(e)$ is equal to

• A. $e\, \log_e\,5$
• B. $e\, \log_e\,3$
• C. $\frac{1}{e\,\log_{e}\,5}$
• D. $\frac{1}{e\,\log_{e}\,3}$

Answer 1

Answer: (C)

We have,
$f(x) =\log _{5} \log _{3} x $
$=\frac{\log \log _{3} x}{\log 5}$
$=\frac{\log \left(\frac{\log x}{\log 3}\right)}{\log 5} $
$=\frac{\log \log x-\log \log 3}{\log 5} $
$\therefore f^{\prime}(x) =\frac{1}{\log 5} \cdot \frac{1}{\log x} \cdot \frac{1}{x}$
$ \therefore f^{\prime}(e) =\frac{1}{e \log 5 \cdot \log e} $
$=\frac{1}{e \log 5} [\because \log e=1] $
$=\frac{1}{e \log _{e} 5} $