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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Basics of Mathematics

NDA 2021 · Mathematics · Basics of Mathematics

If _ 10 2, _ 10 (2^x-1), _ 10 (2^x+3) are in AP, then what is x...

If log⁡102,log⁡10\log _{10} 2, \log _{10}\left(2^x-1\right), \log _{10} (2x+3)\left(2^x+3\right) are in AP, then what is xequalto?x equal to?

Options

  1. A0
  2. B1
  3. Clog⁡25\log _2 5
  4. Dlog⁡52\log _5 2

Correct answer

C. log⁡25\log _2 5

Step-by-step solution

∴log⁡102,log⁡10(2x−1),log⁡10(2x+3) are in  A.P. ∴log⁡10(2x−1)=log⁡1022+log⁡10(2x+3)⇒log⁡10(2x−1)2=log⁡10(2x+3)⇒(2x−1)2=2(2x+3)⇒(2x)2−2.2x+1=2.2x+6⇒(2x)2−4.2x−5=0⇒(2x−5)(2x+1)=0∴2x+1≠0∴2x−5=0∴2x=5⇒x=log⁡25\begin{aligned} & \therefore \log _{10^2}, \log _{10}\left(2^x-1\right), \log _{10}\left(2^x+3\right) \text{ are\text{ in } A.P. } \\ & \therefore \log _{10}\left(2^x-1\right)=\log _{10} 2^2+\log _{10}\left(2^x+3\right) \\ & \Rightarrow \log _{10}\left(2^x-1\right)^2=\log _{10}\left(2^x+3\right) \\ & \Rightarrow\left(2^x-1\right)^2=2\left(2^x+3\right) \\ & \Rightarrow\left(2^x\right)^2-2.2^x+1=2.2^x+6 \\ & \Rightarrow\left(2^x\right)^2-4.2^x-5=0 \\ & \Rightarrow\left(2^x-5\right)\left(2^x+1\right)=0 \\ & \therefore 2^x+1 \neq 0 \\ & \therefore 2^x-5=0 \\ & \therefore 2^x=5 \\ & \Rightarrow x=\log _2 5\end{aligned}
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