solving a functional equation $f(xy)=f(x)+f(y)-1$ on positive integers using given values

Question

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A Function $f$ from the positive integers to the positive integers satisfies the following conditions:

1. $f(xy)=f(x)+f(y)-1$
2. $f(x)=1$ holds for only finitely many $x$.
3. $f(30)=4$

What is the value of $f(14400)$?

This is how I proceeded. Putting $x=y=0$, we get $f(0)=1$ and similarly $f(1)=1$

Since $14400= (144)(100), f(14400)=f(100)+f(144)-1$.

From $f(30)=4$, we get $f(10)+f(3)=5$. Using this and from the above splitting we get $f(10)+f(10)-1+f(3)+f(48)-1=6+f(16)$. I am unable to find $f(16)$.