Question

Let $ f(x) $ be defined for all $ x>0 $ and be continuous, let $ f(x) $ satisfy $ f\left( \frac{x}{y} \right)=f(x)-f(y) $ for all $ x,\,\,y, $ then:

• A. $ f(x)=\log x $
• B. $ f(x) $ is bounded
• C. $ f\left( \frac{1}{2} \right)\to 0 $ as $ x\to 0 $
• D. $ x\,\,f(x)\to 1 $ as $ x\to 0 $

Answer 1

Answer: (A)

Let $f(x)=\log (x), x>0 $
$\therefore $ It is continuous for every positive value of $x$ .
$ \therefore $ $f\left(\frac{x}{y}\right)=\log \left(\frac{x}{y}\right)$
$=\log (x)-\log (y)=f(x)-f(y) $
$\therefore $ Option is correct.