6. If real numbers $x>y>z>t>0$, to make $\log _{\frac{x}{y}} 2009+\log _{\frac{x}{z}} 2009+\log _{\frac{x}{4}} 2009 \geqslant k \log _{\frac{x}{4}} 2009$ always hold, then the maximum value of $k$ is $\qquad$.
6. 9 Detailed Explanation: From the given, $k \leqslant \frac{\log _{\frac{x}{y}} 2009+\log _{\frac{x}{z}} 2009+\log _{\frac{x}{t}} 2009}{\log _{\frac{x}{t}} 2009}=f(x, y, z, t)$.

Since $f(x, y, z, t)=\left(\frac{1}{\log _{2009} \frac{x}{y}}+\frac{1}{\log _{2009} \frac{y}{z}}+\frac{1}{\log _{2009} \frac{z}{t}}\right)\left(\log _{2009} \frac{x}{y}+\log _{2009} \frac{y}{z}+\log _{2009} \frac{z}{t}\right) \geqslant 9$. Therefore, $k_{\max }=9$.