Question

If $f(x)={\log }_5{\log }_{3}x$, then $f^{\prime }(e)$ 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,
$f(x)={\log }_5{\log }_{3}x$
$=\frac{\log {\log }_{3}x}{\log 5}$
$=\frac{\log \left(\frac{\log x}{\log 3}\right)}{\log 5}$
$=\frac{\log \log x-\log \log 3}{\log 5}$
$\therefore f^{\prime }(x)=\frac{1}{\log 5}\cdot \frac{1}{\log x}\cdot \frac{1}{x}$
$\therefore f^{\prime }(e)=\frac{1}{e\log 5\cdot \log e}$
$=\frac{1}{e\log 5}[\because \log e=1]$
$=\frac{1}{e{\log }_{e}5}$