Basics & LogarithmsmediumPYQ · JEE Main · 26 Feb 2021 · Shift 1 (Morning)Free

Basics & Logarithms: Number Solutions Equation (JEE Main 2021)

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

Question
The number of solutions of the equation log⁡4(x−1)=log⁡2(x−3)\log_{4}(x-1)=\log_{2}(x-3) is
Solution
Answer: 1
Step 1: Use log⁡4(x−1)=12log⁡2(x−1)\log_{4}(x-1)=\dfrac{1}{2}\log_{2}(x-1):
12log⁡2(x−1)=log⁡2(x−3)\dfrac{1}{2}\log_{2}(x-1)=\log_{2}(x-3)
log⁡2(x−1)=2log⁡2(x−3)=log⁡2(x−3)2\log_{2}(x-1)=2\log_{2}(x-3)=\log_{2}(x-3)^{2}
 Step 2: So:
x−1=(x−3)2  ⇒  x−1=x2−6x+9x-1=(x-3)^{2}\;\Rightarrow\;x-1=x^{2}-6x+9
x2−7x+10=0  ⇒  (x−2)(x−5)=0x^{2}-7x+10=0\;\Rightarrow\;(x-2)(x-5)=0
 Step 3: Check domain: need x−1>0x-1>0 (so x>1x>1) and x−3>0x-3>0 (so x>3x>3). Combined: x>3x>3. x=2x=2: not >3>3. Rejected. x=5x=5: >3>3. Accepted. Step 4: Only 11 solution. Answer: 11
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