Question

Which of the following denotes the greatest positive proper fraction?

• A. ${\left(\begin{array}{c}\frac{1}{4}\end{array}\right)}^{{\log }_{2}6}$
• B. ${\left(\begin{array}{c}\frac{1}{3}\end{array}\right)}^{{\log }_{3}5}$
• C. $3^{-{\log }_{3}2}$
• D. $8^{\left(\begin{array}{c}\frac{1}{-{\log }_{3}2}\end{array}\right)}$

Answer 1

Answer: (C)

$3^{-{\log }_{3}2}$

A. ${\left(\begin{array}{c}\frac{1}{4}\end{array}\right)}^{{\log }_{2}6}=2^{-2{\log }_{2}6}=\frac{1}{36}$
B. ${\left(\begin{array}{c}\frac{1}{3}\end{array}\right)}^{{\log }_{3}5}=3^{-{\log }_{3}5}=\frac{1}{5}$
C. $3^{-{\log }_{3}2}=\frac{1}{2}$
D. $8^{\left(\begin{array}{c}\frac{1}{-{\log }_{3}2}\end{array}\right)}=2^{-3{\log }_{2}3}=2^{{\log }_2{3}^{-3}}=\frac{1}{27}$
Hence, option 'C' is correct.