Question

If ${\log }_{10}2,{\log }_{10}(2^x-1)$ and ${\log }_{10}(2^x+3)$ are three consecutive terms of an $AP$, then the value of $x$ is

• A. $1$
• B. ${\log }_{5}2$
• C. ${\log }_{2}5$
• D. ${\log }_{10}5$

Answer 1

Answer: (C)

${\log }_{2}5$

If $a,b,c$ are in AP then $2b=a+c$

$\Rightarrow 2{\log }_{10}{(2}^x-1)={\log }_{10}2+{\log }_{10}{(2}^x+3)$

$\Rightarrow {\log }_{10}{{(2}^x-1)}^2={\log }_{10}2(2^x+3)$

$\Rightarrow {\log }_{10}{(2}^{2x}+1-2^{x+1})={\log }_{10}{(2}^{x+1}+6)$

$\Rightarrow 2^{2x}+1-2^{x+1}=2^{x+1}+6$

$\Rightarrow 2^{2x}-2^{x+2}-5=0$

Take, $2^x=y$

$\Rightarrow y^2-4y-5=0$

$\Rightarrow (y-5)(y+1)=0$

$\Rightarrow y=5,-1$

$\because y>0\Rightarrow y=5$

$\Rightarrow 2^x=5$

$\Rightarrow x={\log }_{2}5$