Question
Let $$f\left( x \right)$$ be defined for all $$x > 0$$ and be continues. Let $$f\left( x \right)$$ satisfy $$f\left( {\frac{x}{y}} \right) = f\left( x \right) - f\left( y \right)$$ for all $${x,y}$$ and $$f\left( e \right) = 1.$$ Then
A.
$$f\left( x \right)$$ is bounded
B.
$$f\left( {\frac{1}{x}} \right) \to 0\,{\text{as}}\,x \to 0$$
C.
$$x\,f\left( x \right) \to 1\,{\text{as}}\,x \to 0$$
D.
$$f\left( x \right) = \ln x$$
Answer :
$$f\left( x \right) = \ln x$$
Solution :
$$f\left( x \right)$$ is continuous and defined for all $$x > 0$$ and $$f\left( {\frac{x}{y}} \right) = f\left( x \right) - f\left( y \right)$$
Also $$f\left( e \right) = 1$$
$$ \Rightarrow $$ Clearly $$f\left( x \right) = \ell n\,x$$ which satisfies all these properties
$$\therefore f\left( x \right) = \ell n\,x$$