Question

If $a={\log }_{3}5$ and $b={\log }_{7}25$ then correct option is:

• A. $a<b$
• B. $a>b$
• C. $a=b$
• D. None of these

Answer 1

The correct option is A $a<b$

$a={\log }_{3}5=\frac{\log 5}{\log 3},b=\frac{\log 25}{\log 7}=\frac{\log 5^2}{\log 7}=\frac{2\log 5}{\log 7}$

$\frac{a}{b}=\frac{\log 5}{\log 3}\times \frac{\log 7}{2\log 5}=\frac{1}{2}{\log }_{3}7={\log }_3\left(\sqrt{7}\right)$

$Now,\sqrt{7}<3,so\frac{a}{b}<1$

$\frac{a}{b}<1=>a<b$