Question

If $ f(x) = log_x $ { $ ln(x) $ },then $ f'(e) $ is equal to

• A. $ 1/e $
• B. $ e $
• C. $ -e $
• D. $ e^2 $

Answer 1

Answer: (A)

Given, $f(x)=\log _{x}\{\log (x)\}$
$=\frac{\log \log x}{\log x} \left(\log _{a} b=\frac{\log b}{\log a}\right)$
On differentiating w.r.t. ' $x$ ', we get
$f^{\prime}(x) =\frac{\log x\left[\frac{1}{\log x} \times \frac{1}{x}-\log \log x \times \frac{1}{x}\right]}{(\log x)^{2}} $
$=\frac{\left[\frac{1}{x \log x}-\frac{\log \log x}{x}\right]}{\log x}$
At $x=e$
$f^{\prime}(e) =\frac{\frac{1}{e \log e}-\frac{\log \log e}{e}}{\log e} $
$=\frac{\frac{1}{e}-0}{1}=\frac{1}{e}(\log \log e=\log 1=0)$