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Solve {_2 x^{4}}+4_4{{2}{x}}=2

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Solve $lo g_{2} x^{4} + 4 lo g_{4} \frac{2}{x} = 2$

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lo g_{2} x^{4} + 4 lo g_{4} \frac{2}{x} = 2

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Text solutionVerified

lo g_{2} x^{4} + 4 lo g_{4} \frac{2}{x} = 2

\Rightarrow 4 lo g_{2} x + 4 lo g_{4} 2 - 4 lo g_{4} x = 2

\Rightarrow 2 lo g_{2} x + 4 lo g_{4} 2 - 4 lo g_{4} x = 2

\Rightarrow lo g_{2} x + 2 lo g_{4} 2 - 2 lo g_{4} x = 1

\Rightarrow lo g_{2} x + \frac{1}{2} lo g_{2} 2 - \frac{2}{2} lo g_{2} x = 1

\Rightarrow lo g_{2} x + \frac{1}{2} - \frac{2}{2} lo g_{2} x = 1

\Rightarrow lo g_{2} x - \frac{2}{2} lo g_{2} x = 1 - \frac{1}{2}

\Rightarrow lo g_{2} x - lo g_{2} x = \frac{1}{2}

Let

α = lo g_{2} x \Rightarrow α^{2} = lo g_{2} x

\Rightarrow α - α^{2} = \frac{1}{2}

\Rightarrow 2 α - 2 α^{2} - 1 = 0

\Rightarrow 2 α^{2} - 2 α - 1 = 0

\Rightarrow α = \frac{2 \pm 4 + 8}{4} = \frac{2 \pm 2 3}{4} = \frac{\pm 3}{2}

\Rightarrow lo g_{2} x = \frac{1 + 3}{2} ∵ x > 0

\Rightarrow lo g_{2} x = (\frac{1 + 3}{2})^{2} = \frac{1 + 3 + 2 3}{4} = \frac{2 + 3}{2}

∴ x = 2^{\frac{2 + 3}{2}}

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Question Text	Solve $lo g_{2} x^{4} + 4 lo g_{4} \frac{2}{x} = 2$
Answer Type	Text solution:1
Upvotes	150