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JEE MainMathematicsModerateLogarithm and its applications

The number of integral solutions of (log_.5x)^2 +
log_.5x−2≤0- 2 \leq 0 is

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(log⁡0.5 x)2+(log⁡0.5x)−2≤0\left( \log_{0.5}\ x \right)^{2} + \left( \log_{0.5}x \right) - 2 \leq 0
Put - log⁡0.5−x=1\log_{0.5} - x = 1
⇒\Rightarrow u2+t−2≤0u^{2} + t - 2 \leq 0
⇒ (1t+2)(t−1)≤0(1t + 2)(t - 1) \leq 0
−2≤u≤1- 2 \leq u \leq 1
−2≤log⁡0.5n≤1- 2 \leq \log_{0.5}n \leq 1
(0.5)−2≥x≥0.5(0.5)^{- 2} \geq x \geq 0.5
12≤x≤4\frac{1}{2} \leq x \leq 4
Integral solution = 1, 2, 3, 4

Source: Grass-G / M-05 Mathematics Paper 10 October.docx

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