I'm working through some homework on induction, and most problems I can solve fine, but I have problem getting started on induction proofs that ask you to prove function relations. For example, here is one problem from my textbook (not homework), that I don't know how to solve:
Suppose a function $f(x)$ has the property that $f(x_1x_2)$ = $f(x_1) + f(x_2)$ for any two positive numbers $x_1$ and $x_2$. Show that $f(x_1x_2···x_n)$ = $f(x_1) + f(x_2) + · · · + f(x_n)$ for the product of any $n$ positive numbers $x_1, x_2, · · · , x_n$.
I can solve problems asking to do 'arithmetic' induction, such as "Prove $n^2 > 4n + 1$ for integers n ≥ 5", but any of these problems relating to functions I can't seem to wrap my head around. What do you differently on these type of problems or how do you do induction on them?