Question

If $4^{{\log }_{16}4}+9^{{\log }_{3}9}={10}^{{\log }_{x}83}$, then the value of $x$ is

Answer 1

${10}^{{\log }_{x}83}=4^{{\log }_{16}4}+9^{{\log }_{3}9}=4^{{\log }_{2^4}{2}^2}+9^{{\log }_3{3}^2}$
$\Rightarrow {10}^{{\log }_{x}83}=4^{\frac{2}{4}{\log }_{2}2}+9^{2{\log }_{3}3}$

$[\because {\log }_{a^n}{b}^m=\frac{m}{n}{\log }_{a}b]$

$=4^{1/2}+9^2=2+81$

$\Rightarrow {10}^{{\log }_{x}83}={83}^{{\log }_{x}10}[\because a^{{\log }_{b}c}=c^{{\log }_{b}a}]={83}^1$

Equating power of $10$
$\Rightarrow x=10$
Ans:10