Question

The number of real solution(s) of the equation ${\log }_4(1+x^4{)}^2=2-{\log }_2(4+x^2)$ is

Answer 1

${\log }_4(1+x^4{)}^2=2-{\log }_2(4+x^2)\Rightarrow {\log }_{2^2}(1+x^4{)}^2=2-{\log }_2(4+x^2)$
$\Rightarrow {\log }_2(1+x^4)=2-{\log }_2(4+x^2)$
$\Rightarrow {\log }_2(1+x^4)+{\log }_2(4+x^2)=2$
$\Rightarrow {\log }_2\left(\right(1+x^4\left)\right(4+x^2\left)\right)=2\Rightarrow (1+x^4)(4+x^2)=4$

We know that,
$1+x^4\ge 14+x^2\ge 4\Rightarrow (1+x^4)(4+x^2)\ge 4$
So,
$(1+x^4)(4+x^2)=4\text{when}x=0$