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Fundamental Laws of Logarithms

The number of...

Question

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

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Solution

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

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

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