1. Let $a_{1}, a_{2}, \cdots, a_{10} \in(1,+\infty)$. Then
$$
\frac{\log _{a_{1}} 2009+\log _{a_{2}} 2009+\cdots+\log _{a_{10}} 2009}{\log _{a_{1,2} \cdots a_{10}} 2009}
$$

the minimum value is
-1.100 .
$$
\begin{array}{l}
\text { Original expression }=\left(\sum_{i=1}^{10} \frac{\lg 2009}{\lg a_{i}}\right) \frac{\lg \left(\prod_{i=1}^{10} a_{i}\right)}{\lg 2009} \\
=\left(\sum_{i=1}^{10} \frac{1}{\lg a_{i}}\right)\left(\sum_{i=1}^{10} \lg a_{i}\right) \\
\geqslant 10^{2},
\end{array}
$$

Equality holds if and only if $a_{1}=a_{2}=\cdots=a_{10}$. Therefore, the minimum value of the original expression is 100.