14. $f(x)$ is a function defined on $(0,+\infty)$ that is increasing, and $f\left(\frac{x}{y}\right)=f(x)-f(y)$.
(1) Find the value of $f(1)$; (2) If $f(6)=1$, solve the inequality $f(x+3)-f\left(\frac{1}{x}\right)<2$.
14. (1) Let $x=y=1$, we get $f(1)=0$;
(2) The original inequality is transformed into $\left\{\begin{array}{l}f(x(x+3))<0 \\ \frac{1}{x}>0 \\ \frac{x(x+3)}{6}<6,\end{array}\right.$ which gives $0<x<\frac{-3+3 \sqrt{17}}{2}$.