Solution:
Applying laws of logarithms to the given equation, we get
$$\log_{250}\left(2^{a} 5^{b}\right) = 3 \text{ or } \quad 2^{a} 5^{b} = 250^{3} = 2^{3} 5^{9}$$
Since $a$ and $b$ are integers and $\gcd(2,5) = 1$, we get $a = 3$ and $b = 9$, so that $a + 2b = 21$.