Q151 / 270 · NDA 2017 (Phase 2) · Mathematics

If $x+\log _{10}\left(1+2^x\right)=x \log _{10} 5+\log _{10} 6$ then $x$ is equal to

  • A
    $2,-3$
  • B
    $2$ only
  • Correct answer
    C
    $1$
  • D
    $3$

Solution

$\begin{aligned} & x+\log _{10}\left(1+2^x\right)=x \log _{10} 5+\log _{10} 6 \\ & \Rightarrow x-x \log _{10} 5=\log _{10} 6-\log _{10}\left(1+2^x\right) \\ & \Rightarrow x\left(1-\log _{10} 5\right)=\log _{10} 6-\log _{10}\left(1+2^x\right) \\ & \Rightarrow x\left(\log _{10} 10-\log _{10} 5\right)=\log _{10}\left(\frac{6}{1+2^x}\right) \\ & \Rightarrow x\left(\log _{10}\left(\frac{10}{5}\right)\right)=\log _{10}\left(\frac{6}{1+2^x}\right) \\ & \Rightarrow x \log _{10} 2=\log _{10}\left(\frac{6}{1+2^x}\right)\end{aligned}$ This is possible only when $x=1$.

Hence, the correct answer is option: 'C'.

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