Question

If $4^{{\log }_{9}3}+9^{{\log }_{2}4}={10}^{{\log }_{x}83},$ then $x$ is equal to:

• A. $2$
• B. $10$
• C. $9$
• D. None of the above

Answer 1

The correct option is B $10$

$4^{{\log }_{9}3}+9^{{\log }_{2}4}={10}^{lo{g}_{x}83}$

$4^{{\log }_{3^2}3}+9^{{\log }_{2}4}={10}^{lo{g}_{x}83}$

$\Rightarrow 4^{1/2{\log }_{3}3}+9^{2{\log }_{2}2}={10}^{{\log }_{x}83}\dots ({\log }_{b^x}a=\frac{1}{x}{\log }_{b}a)$

$\Rightarrow 4^{1/2}+(9{)}^2=(10{)}^{lo{g}_x\left(83\right)}$

$\Rightarrow (2+81)=(10{)}^{{\log }_{x}83}$

$\Rightarrow \left(83\right)={10}^{log\times 83}$

$\Rightarrow {\log }_{10}83={\log }_{x}83$

$\Rightarrow x=10$