The functional equation $f(xy)=f(x)f(y)$

Let $f(x)$ be a function that satisfies this functional equation, $f(xy)=f(x)f(y)$.

With a little bit of intuition and luck one may come to a conclusion that these are perhaps the solutions of $f(x)$,

• $f(x)=x$


• $f(x)=1$


• $f(x)=0$

However, these solutions are family solutions of $f(x)=x^n$. What I meant by this is that, when $n=1$ you get the function $f(x)=x$. When $n=0$ you get $f(x)=1$ and when $x=0$ well you get $f(x)=0$.
So, it seems $f(x)=x^n$ is the genuine solution to that functional equation and when you're taking different values for $x$ and $n$ you're getting bunch of other functions of the same family.

Getting excited by this I tried to take different values for $x$, for instance when $x=2$, $x^n$ becomes $2^n$. So, now I'm expecting the function $f(x)=2^n$ to satisfy this functional equation $f(xy)=f(x)f(y)$. However, it doesn't. I don't know why it's not satisfying. May I get your explanation?