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JEE Main 2001 Mathematics question on Complex Numbers and Quadratic Equations

The number of solutions of log⁡4(x−1)=log⁡2(x−3)\log_4(x-1) = \log_2(x-3)
  1. A.
    3
  2. B.
    1
  3. C.
    2
  4. D.
    0
Show answer and solution

Correct option: B

Solution

log₄(x-1) = log₂(x-3) can be rewritten as log₂(x-1)/2 = log₂(x-3), so log₂(x-1) = 2·log₂(x-3) = log₂(x-3)². This gives x-1 = (x-3)², i.e., x²-7x+10 = 0, so x = 5 or x = 2. Since the domain requires x-1 > 0 and x-3 > 0, we need x > 3. Only x = 5 satisfies this, giving exactly 1 solution.

[2001]

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