Question

Find the value of: $N=\frac{{\log }_{5}250}{{\log }_{50}5}-\frac{{\log }_{5}10}{{\log }_{1250}5}$.

Answer 1

$N=\frac{{\log }_{5}250}{{\log }_{50}5}-\frac{{\log }_{5}10}{{\log }_{1250}5}={\log }_{5}250{\log }_{5}50-{\log }_{5}1250{\log }_{5}10.................[\because {\log }_{a}b=\frac{\log b}{\log a}]$

$\Rightarrow N=({\log }_{5}10+2{\log }_{5}5)({\log }_{5}10+{\log }_{5}5)-({\log }_{5}10+3{\log }_{5}5){\log }_{5}10$

$[\because \log (ab)$$=\log a+\log b\text{\&}\log a^m=m\log a]$

$\Rightarrow N=2+3{\log }_{5}10-3{\log }_{5}10=2$

$\because [{\log }_{a}a=1]$
Ans: 2