$\text{ The value of }N=\frac{{\log }_{5}250}{{\log }_{50}5}-\frac{{\log }_{5}10}{{\log }_{1250}5}\text{ is }$

$\begin{array}{l}\mathrm{N}=\frac{{\log }_{5}250}{{\log }_{50}5}-\frac{{\log }_{5}10}{{\log }_{1250}5}\\ =\left({\log }_{5}250\right)\left({\log }_{5}50\right)-\left({\log }_{5}10\right)\left({\log }_{5}1250\right)\\ =\left(3+{\log }_{5}2\right)\left(2+{\log }_{5}2\right)-\left(1+{\log }_{5}2\right)\left(4+{\log }_{5}2\right)\\ ={\left({\log }_{5}2\right)}^2+5{\log }_{5}2+6-\left[{\left({\log }_{5}2\right)}^2+5{\log }_{5}2+4\right]\\ =2\end{array}$