$f$ is a continuous real-valued function such that $f(x+y)=f(x) f(y)$ for all real $x, y$. If $f(2)=5$, find $f(5)$.
Solution 1: Since $f(n x)=f(x)^{n}$ for all integers $n, f(5)=f(1)^{5}$ and $f(2)=f(1)^{2}$, so $f(5)=$ $f(2)^{5 / 2}=\mathbf{2 5} \sqrt{\mathbf{5}}$.

Solution 2: More generally, since $f(n x)=f(x)^{n}$ for all integers $n, f(1)=c=f(1 / n)^{n}$ for some constant $c$ and all integers $n$. Thus $f(k / n)=f(1 / n)^{k}=f(1)^{k / n}=c^{k / n}$ for all rational numbers $k / n$. By continuity, it follows that $f(x)=c^{x}$ for all real numbers $x$. Since $f(2)=5, c=\sqrt{5}$, so $f(5)=25 \sqrt{5}$