Problem #12: Almost multiplicative

Let \(f\) be a function such that \(f(210) = 5\) and

$$\forall n, m \in \mathbb N, \; f(nm) = f(n) + f(m) - 1$$

Find the value of \(f(2^{10}\cdot 3^8 \cdot 5^6 \cdot 7^4)\).