Question

If $x={\log }_5\left(1000\right)$ and $y={\log }_7\left(2058\right)$, then relation between $x$ and $y$ is

• A. $x>y$
• B. $x>=y$
• C. $x<=y$
• D. $x<y$

Answer 1

Answer: (A)

$x>y$

$x={\log }_{5}1000={\log }_5{5}^3\times 8=3{\log }_{5}5+{\log }_{5}8=3+{\log }_{5}8$

$[\because \log (ab)=\log a+\log b]=3+{\log }_{5}8$

$y={\log }_{7}2058={\log }_7(7^3\cdot 6)[\because \log (ab)=\log a+\log b]=3+{\log }_{7}6$

${\log }_{5}8>{\log }_{5}5\Rightarrow {\log }_{5}8>1\therefore x>4$

${\log }_{7}6<{\log }_{7}7\Rightarrow {\log }_{7}6<1\therefore y<4$

$\therefore x>y$