(Example 10 Given a function $f(x)$ defined on $\mathbf{R}^{+}$, for any positive numbers $x_{1}$ and $x_{2}$, it holds that $f\left(x_{1} x_{2}\right)=$ $f\left(x_{1}\right)+f\left(x_{2}\right)$, and $f(8)=3$. Find the value of $f(\sqrt{2})$.