We are to find a function f which follows the following properties $$f(mn)=f(m)f(n),\; f(2)=2.$$
I can easily find all the values of $f(2^n)$ but I am confused on how to find for the odd numbers and the primes so as to find the function using induction. Please help.
Let $g:\mathbb{P}\rightarrow \mathbb{R}$ be a given function with $g(2)=2$, in which $\mathbb{P}$ is set of all prime number. Define the function $f:\mathbb{N}\rightarrow \mathbb{R}$ as follow $$f(n)=\prod_{i=1}^{k}g(p_{i})^{s_i}$$ where $n=\prod_{i=1}^{k}p_{i}^{s_i}$ factorises into distinct prime powers. Then $$f(mn)=f(n)f(m)$$ and $f(2)=2$. Therefore there is infinitly functions that satisfy in your functional equation.
Hint. This functional equation is called "Multiplicative Arithmetic Functions" and is an important branch of Theory of Arithmetic Functions and also there is a lot of results about this subject.
R Sivaramakrishnan, Classical Theory of Arithmetic Functions, CRC Press