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

Question

$\begin{array}{1 1} f(x) \;is\;bounded \\ f(\frac{1]{x})=>0\;and\;x=0 \\ xf(x)=>1\;as\;x=0 \\f(x) =\log x \end{array}$