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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Basics of Mathematics

NDA 2024 · Mathematics · Basics of Mathematics

What is the number of solutions of _4(x-1)= _2(x-3) ?

What is the number of solutions of log⁡4(x−1)=log⁡2(x−3)\log _4(x-1)=\log _2(x-3) ?

Options

  1. AZero
  2. BOne
  3. CTwo
  4. DThree

Correct answer

B. One

Step-by-step solution

log⁡4(x−1)=log⁡2(x−3)\log _4(x-1)=\log _2(x-3)⇒12log⁡(x−1)=log⁡2(x−3)⇒log⁡2(x−1)=log⁡2(x−3)2⇒x−1=x2−6x+9⇒x2−7x+10=0⇒x2−5x−2x+10=0⇒x(x−5)−2(x−5)=0⇒(x−5)(x−2)=0\begin{aligned} & \Rightarrow \quad \frac{1}{2} \log (x-1)=\log _2(x-3) \\ & \Rightarrow \quad \log _2(x-1)=\log _2(x-3)^2 \\ & \Rightarrow \quad x-1=x^2-6 x+9 \\ & \Rightarrow \quad x^2-7 x+10=0 \\ & \Rightarrow x^2-5 x-2 x+10=0 \\ & \Rightarrow x(x-5)-2(x-5)=0 \\ & \Rightarrow \quad(x-5)(x-2)=0\end{aligned}\left[x≠2 because x−3>0,x>3\begin{array}{l}x \neq 2 \text{ because } \\ x-3 > 0, x > 3\end{array}\right]⇒x=2,5⇒x=5\begin{array}{ll} \Rightarrow & x=2,5 \\ \Rightarrow & x=5 \end{array}Hence,onesolution.Hence, one solution.
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