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Basics & Logarithms: Find Sum Values Satisfying Equation

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

Question
Find the sum of all values of xx satisfying the equation log⁡4(x−1)=log⁡2(x−3)\log_4(x-1) = \log_2(x-3) is
Solution
Answer: 5
Step 1: Convert log⁡4(x−1)\log_4(x-1) to base 2:
log⁡2(x−1)2=log⁡2(x−3)  ⟹  log⁡2(x−1)=2log⁡2(x−3)=log⁡2(x−3)2\frac{\log_2(x-1)}{2} = \log_2(x-3) \implies \log_2(x-1) = 2\log_2(x-3) = \log_2(x-3)^2
 Step 2: Therefore x−1=(x−3)2=x2−6x+9x-1 = (x-3)^2 = x^2-6x+9.
x2−7x+10=0  ⟹  (x−5)(x−2)=0  ⟹  x=5 or x=2x^2-7x+10=0 \implies (x-5)(x-2)=0 \implies x=5 \text{ or } x=2
 Step 3: Check domain: need x−1>0x-1>0 and x−3>0x-3>0, so x>3x>3. x=5>3x=5 > 3: valid. x=2x=2: x−3=−1<0x-3=-1<0, invalid (logarithm undefined). Rejected. Only solution is x=5x=5. Sum =5= 5. Answer: 5
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