Question

Which of the following statements is/are not true?

• A. ${\log }_{2}3<{\log }_{12}10$
• B. ${\log }_{6}5<{\log }_{7}8$
• C. ${\log }_{3}26<{\log }_{2}9$
• D. ${\log }_{16}15<{\log }_{10}11>{\log }_{7}6$

Answer 1

The correct option is A ${\log }_{2}3<{\log }_{12}10$

$A)$ ${\log }_{2}3<{\log }_{12}10$

${\log }_{2}3=\frac{\log 3}{\log 2}$ as $\log 3>\log 2\Rightarrow \frac{\log 3}{\log 2}>1$

and ${\log }_{12}10=\frac{\log 10}{\log 12}$

as $\log 12>\log 10\Rightarrow \frac{\log 10}{\log 12}<1$

Hence, ${\log }_{2}3\nless {\log }_{12}10$

$B)$ ${\log }_{6}5<{\log }_{7}8$

${\log }_{6}5=\frac{\log 5}{\log 6}$ as $\log 6>\log 5\Rightarrow {\log }_{6}5<1$

${\log }_{7}8=\frac{\log 8}{\log 7}$ as $\log 7<\log 8\Rightarrow {\log }_{7}8>1$

Hence, ${\log }_{6}5<{\log }_{7}8$

$C)$ ${\log }_{3}26<{\log }_{2}9$

${\log }_{2}9=3.17{\log }_{3}26=2.96$

Hence, ${\log }_{3}26<{\log }_{2}9$

$D)$ ${\log }_{16}15<{\log }_{10}11>{\log }_{7}6$

${\log }_{16}15=\frac{\log 15}{\log 16}$ as $\log 16>\log 15\Rightarrow {\log }_{16}15<1$

${\log }_{10}11=\frac{\log 11}{\log 10}$ as $\log 11>\log 10\Rightarrow {\log }_{10}11>1$

${\log }_{7}6=\frac{\log 6}{\log 7}$ as $\log 7>\log 6\Rightarrow {\log }_{7}6<1$

Hence, ${\log }_{16}15<{\log }_{10}11>{\log }_{7}6$