Question

Let $f\left(x\right)$ be defined for all $x>0$ and be continuous,Let $f\left(x\right)$ satisfy $f\left(\frac{x}{y}\right)=f\left(x\right)-f\left(y\right)$ for all x,y, $f\left(e\right)=1$ Then

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

Answer 1

Answer: (D)

$f\left(x\right)=\log x$

Since,$f\left(\frac{x}{y}\right)=f\left(x\right)-f\left(y\right)$ for all x,y
Clearly,the function$f\left(x\right)=a\log x,$where a is any real number.
Since,$f\left(e\right)=a\log e=1\Rightarrow a=1$,
$f\left(x\right)=\log x$
Clearly,$f$is not bounded,
As $x\to 0,f\left(\frac{1}{x}\right)\to -\infty$ and $xf\left(x\right)\to 0$
So, the correct option is (D).