Question

The value of ${49}^{(1-{\log }_{7}2)}+5^{-{\log }_{5}4}$ is equal to

• A. $\frac{27}{2}$
• B. $\frac{25}{2}$
• C. $\frac{625}{16}$
• D. none of these

Answer 1

The correct option is B $\frac{25}{2}$

${49}^{(1-{\log }_{7}2)}+5^{-{\log }_{5}4}$
$={49}^{\left({\log }_{7}7-{\log }_{7}2\right)}+5^{{\log }_5{4}^{-1}},$ $(\because$ ${\log }_{a}a=1)$
$={49}^{{\log }_7\frac{7}{2}}+4^{-1},(\because a^{{\log }_{a}y}=y)$
$=7^{2{\log }_7\frac{7}{2}}+\frac{1}{4}$
$=7^{{\log }_7{\left(\frac{7}{2}\right)}^2}+\frac{1}{4}$ $,(\because a^{{\log }_{a}y}=y^{{\log }_{a}a}=y)$
$={\left(\frac{7}{2}\right)}^2+\frac{1}{4}=\frac{50}{4}=\frac{25}{2}$