Question

Statement$-1:{\log }_{10}x<{\log }_{\pi }x<{\log }_{e}x<{\log }_{2}x\forall x>1$
Statement$-2:x<y\Rightarrow {\log }_{a}x>{\log }_{a}y$ when $0<a<1$

• A. Statement$-1$ is true, Statement$-2$ is true, Statement$-2$ is a correct explanation of Statement$-1$.
• B. Statement$-1$ is true, Statement$-2$ is true, Statement$-2$ is not a correct explanation of Statement$-1$.
• C. Statement$-1$ is true, Statement$-2$ is false.
• D. Statement$-1$ is false, Statement$-2$ is true.

Answer 1

The correct option is B Statement$-1$ is true, Statement$-2$ is true, Statement$-2$ is not a correct explanation of Statement$-1$.

If $\text{base}>1$, then $\log$ function is strictly increasing.
So, when $x>1$, we have
${\log }_{x}10>{\log }_x\pi >{\log }_{x}e>{\log }_{x}2$
$\Rightarrow \frac{1}{{\log }_{x}10}<\frac{1}{{\log }_x\pi }<\frac{1}{{\log }_{x}e}<\frac{1}{{\log }_{x}2}$
$\Rightarrow {\log }_{10}x<{\log }_{\pi }x<{\log }_{e}x<{\log }_{2}x$

For $0<a<1$, ${\log }_{a}x$ is strictly decreasing function of $x$.
$\therefore x<y\Rightarrow {\log }_{a}x>{\log }_{a}y$