Question

If $4^{{\log }_{9}3}+9^{{\log }_{2}4}={10}^{{\log }_{x}83}$, then the value of $x$ is

• A. $10$
• B. $4$
• C. $-10$
• D. $-4$

Answer 1

The correct option is A $10$

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

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

$\Rightarrow 4^{\frac{1}{2}{\log }_{3}3}+9^{2{\log }_{2}2}={10}^{{\log }_{x}83}$

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

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

Applying log on both sides

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

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

$\Rightarrow {\log }_{10}83={\log }_{x}83$ ...Since ${\log }_{10}10=1$

$\Rightarrow x=10$ ....On comparison