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. $xf(x)\to 1$ 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.