Question

Find the value of the following expression: ${81}^{\frac{1}{{\log }_{5}3}}+{27}^{{\log }_{9}36}+3^{\frac{4}{{\log }_{7}9}}$

Answer 1

let $y={81}^{\frac{1}{{\log }_{5}3}}+{27}^{{\log }_{9}36}+3^{\frac{4}{{\log }_{7}9}}={\left(3^4\right)}^{{\log }_{3}5}+{\left(3^3\right)}^{{\log }_{3^2}\left(6^2\right)}+3^{4{\log }_{9}7}[\because {\log }_{a}b=\frac{1}{{\log }_{b}a}]$
$\Rightarrow y=(3{)}^{{\log }_3{5}^4}+{\left(3^3\right)}^{{\log }_3\left(6\right)}+(3{)}^{4{\log }_{3^2}7}[\because m\log a=\log a^m]$
$\Rightarrow y=5^4+3^{{\log }_3{6}^3}+3^{2{\log }_{3}7}[\because {\log }_{a^m}b=\frac{1}{m}{\log }_{a}b]$
$\Rightarrow y=5^4+6^3+3^{{\log }_3{7}^2}=625+216+7^2=890$
Ans: 890