Question

The number of values of $x$ satisfying the equation $2^{\left(\log _5 16\right)\left(\log _4 x\right)+\left(\log _{\sqrt[x]{2}} 5\right)}+x^{\left(\log _5 4\right)+5}+x^5=0$ is

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (A)

$ 2^{\log _5 16 \cdot \log _4 x+\log _x \sqrt{2}^5}+5^x+x^{\log _3 4+5}+x^5=0 $
$2^{2 \log _5 4 \cdot \log _4 x+x \log _2 5}+5^x+x^{\log _5 4} \cdot x^5+x^5=0$
$2^{2 \log _5 x} 2^{x \log _2 5}+5^x+x^{2 \log _5 2} \cdot x^5+x^5=0$
$\left(2^{\log _5 x }\right)^2 \cdot 5^{ x }+5^{ x }+\left(2^{\log _5 x }\right)^2 \cdot x ^5+ x ^5=0 $
$5^{ x }\left[\left(2^{\log _5 x }\right)^2+1\right]+ x ^5\left[\left(2^{\log _5 x }\right)^2+1\right]=0 $
$\left(5^{ x }+ x ^5\right)\left[\left(2^{\log _5 x }\right)^2+1\right]=0$
$5^{ x }+ x ^5=0 \left(2^{\log _5 x }\right)^2+1=0 $
$\text { This possible only when } x \text { will be-ve } \text { No solution } $
$\text { while according to question } x \geq 2 $
$\therefore \text { number of values of } x =\text { zero }$