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Basics & Logarithms: Complete Set Values Expression Defined

JEE Maths question with a full step-by-step solution.

Question
The complete set of values of xx for which the expression y=log⁡1/2(x−1x−2)y = \sqrt{\log_{1/2}\left(\dfrac{x-1}{x-2}\right)} is defined:
A(−∞,1)∪(2,∞)(-\infty, 1) \cup (2, \infty)
B(−∞,2)(-\infty, 2)
C(−∞,1)(-\infty, 1)correct
Dϕ\phi
Solution
Step 1: Require the logarithm to be non-negative
log⁡1/2(x−1x−2)≥0\log_{1/2}\left(\frac{x-1}{x-2}\right) \geq 0
 Since base 12<1\dfrac{1}{2} < 1, this gives 0<x−1x−2≤10 < \dfrac{x-1}{x-2} \leq 1. Step 2: Solve the two parts x−1x−2>0\dfrac{x-1}{x-2} > 0: x<1x < 1 or x>2x > 2. x−1x−2≤1\dfrac{x-1}{x-2} \leq 1: 1x−2≤0  ⟹  x<2\dfrac{1}{x-2} \leq 0 \implies x < 2. Step 3: Intersect
x∈(−∞,1)x \in (-\infty, 1)
 Answer: (3)
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