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JEE Advanced
Exam
JEE Advanced
Subject
Mathematics
Chapter
Basics of Mathematics

JEE Advanced · Mathematics · Basics of Mathematics

Find the number of integral values of x satisfying the equation: |...

Find the number of integral values of xx satisfying the equation: ∣log⁡22x−5log⁡2x+6∣=∣log⁡22x−7log⁡2x+12∣−∣2log⁡2x−6∣\left|\log _2^2 x-5 \log _2 x+6\right|=\left|\log _2^2 x-7 \log _2 x+12\right|-\left|2 \log _2 x-6\right|

Correct answer

5

Step-by-step solution

∣log⁡22x−5log⁡2x+6∣+∣2log⁡2x−6∣=∣log⁡22x−7log⁡2x+12∣∣x∣+∣y∣=∣x−y∣⇒xy≤0∴(log⁡22x−5log⁡2x+6)(2log⁡2x−6)≤0(log⁡2x−2)(log⁡2x−3)(log⁡2x−3)≤0(log⁡2x−2)(log⁡2x−3)2≤0⇒log⁡2x≤2 or log⁡2x=3x∈(0,4]∪{8}\begin{aligned} & \left|\log _2^2 x-5 \log _2 x+6\right|+\left|2 \log _2 x-6\right|=\left|\log _2^2 x-7 \log _2 x+12\right| \\ & |x|+|y|=|x-y| \Rightarrow x y \leq 0 \\ & \therefore\left(\log _2^2 x-5 \log _2 x+6\right)\left(2 \log _2 x-6\right) \leq 0 \\ & \left(\log _2 x-2\right)\left(\log _2 x-3\right)\left(\log _2 x-3\right) \leq 0 \\ & \left(\log _2 x-2\right)\left(\log _2 x-3\right)^2 \leq 0 \\ & \Rightarrow \log _2 x \leq 2 \text { or } \log _2 x=3 \\ & \quad x \in(0,4] \cup\{8\}\end{aligned}
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