2021 AMC 10A Problem 18

Let f be a function defined on the set of positive rational numbers with the property that f(a\cdot b)=f(a)+f(b) for all positive rational numbers a and b. Furthermore, suppose that f also has the property that f(p)=p for every prime number p. For which of the following numbers x is f(x)<0?

\textbf{(A)} ~\frac{17}{32}\qquad\textbf{(B)} ~\frac{11}{16}\qquad\textbf{(C)} ~\frac{7}{9}\qquad\textbf{(D)} ~\frac{7}{6} \qquad\textbf{(E)} ~\frac{25}{11}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.