7. If $a$ and $b$ are integers such that $a \log _{250} 2+b \log _{250} 5=3$, what is the value of $a+2 b$ ?
7. 21
Applying laws of logarithms to the given equation, we get
$$
\log _{250}\left(2^{a} 5^{b}\right)=3 \text { or } 2^{a} 5^{b}=250^{3}=2^{3} 5^{9} .
$$

Since $a$ and $b$ are integers and $\operatorname{gcd}(2,5)=1$, we get $a=3$ and $b=9$, so that $a+2 b=21$.