We know that if a function $f:\mathbb{N}_+\to \mathbb{R}$ where $\mathbb{N}_+=\{1,2,\dots,\}$ satisfies $f(xy)=f(x)+f(y)$ for all $x,y\in\mathbb{N}_+$, then $f$ need NOT be of the form $a\log_b x$. For example, we arbitrarily assign a real number $a_p$ for each prime number $p$ and let $f(1):=0$, $f(p):=a_p$ and $f(p_1\dots p_k):=f(p_1)+\dots+f(p_k)$ for all positive intergers $k$ and prime numbers $p_1,\dots,p_k$.
What if we add a condition that $f$ is increasing, i.e. $f(x)>f(y)$ for all $x>y$ ?