Basic functional equation: $ f ( x y ) = f ( x ) + f ( y ) $

Question

Let $ f $ be any function defined on the set $ \mathbb N $. If $ f ( x y ) = f ( x ) + f ( y ) $ and $ f ( 2 ) = 9 $, then find $ f ( 3 ) $.

The answer to this question is $ 7 $. Please tell me how to get it.

Answer 1

If $f$ is continuous, we have $f(x)=\log_a(x)$ for $a$ such that $a^9=2$. So $a>1$ and $f(x)$ is monotonically increasing and we cannot t have $f(3)=7<9=f(2)$.

Answer 2

Your claim is not true. For instance, $f(x)=9(a_{p_1}+\cdots+a_{p_h})$, where $x=p_1^{a_1}\cdots p_h^{a_{p_h}}$ is its prime factorization, satisfies $f(xy)=f(x)+f(y)$ and $f(2)=9$. Yet, $f(3)=9$. Specifically, for all functions $g:\{\text{primes}\}\to \Bbb N$, there is exactly one function $f:\Bbb N\setminus\{0\}\to\Bbb N$ such that $f(p)=g(p)$ for all primes and $f(xy)=f(x)+f(y)$ for all $x,y$.