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NDA 2018
Exam
NDA
Year
2018
Subject
Mathematics
Chapter
Basics of Mathematics

NDA 2018 · Mathematics · Basics of Mathematics

If x+ _ 15 (1+3^x)=x _ 15 5+ _ 15 12 , where x is an integer, then...

If x+log⁡15(1+3x)=xlog⁡155+log⁡1512x+\log _{15}\left(1+3^x\right)=x \log _{15} 5+\log _{15} 12, where xx is an integer, then what is xx equal to?

Options

  1. A−3-3
  2. B22
  3. Cl
  4. D33

Correct answer

C. l

Step-by-step solution

x+log⁡15(1+3x)=xlog⁡155+log⁡1512⇒x⋅log⁡1515+log⁡15(1+3x)=xlog⁡155+log⁡1512(∵log⁡1515=1)⇒log⁡1515x+log⁡15(1+3x)=log⁡155x+log⁡1512⇒log⁡1515x(1+3x)=log⁡15(5x×12)(∵log⁡a+log⁡b=log⁡ab)\begin{aligned} & x+\log _{15}\left(1+3^x\right)=x \log _{15} 5+\log _{15} 12 \\ & \Rightarrow x \cdot \log _{15} 15+\log _{15}\left(1+3^x\right)=x \log _{15} 5+\log _{15} 12 \\ & \left(\because \log _{15} 15=1\right) \\ & \Rightarrow \quad \log _{15} 15^x+\log _{15}\left(1+3^x\right)=\log _{15} 5^x+\log _{15} 12 \\ & \Rightarrow \log _{15} 15^x\left(1+3^x\right)=\log _{15}\left(5^x \times 12\right) \\ & (\because \log a+\log b=\log a b) \end{aligned}⇒15x(1+3x)=5x×12⇒3x(1+3x)=12⇒3x+32x=12\begin{aligned} & \Rightarrow \quad 15^x\left(1+3^x\right)=5^x \times 12 \\ & \Rightarrow \quad 3^x\left(1+3^x\right)=12 \\ & \Rightarrow \quad 3^x+3^{2 x}=12 \end{aligned}\mathrm{x}=1$ satisfies the above equation.
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