Given $f(xy)=f(x)+f(y)$ such that $f(8)=15$ and $f(12)=24$ can we find explicitly $f(x)$
From $$f(8)=f(4)+f(2)$$ and $$f(4)=2f(2)$$
So $$f(2)=5$$
Now the functional equation is actually satisfied by $$f(x)=\log_{a} x$$ If we choose base $a$ as fifth root of $2$ we have $$f(x)=\log_{2^{.2}} x$$ But with this $f(x)$ it wont satisfy $f(12)=24$ So can we find $f(x)$ explicitly
Your current rules very much underdetermine the function. Infinitely many functions satisfy what you have listed so far. You need to add more to specify a particular function. So no, you cannot find $f(x)$. Examples of how to construct multiple functions satisfying this are in the comments.