Question

If ${\log }_{10}2,{\log }_{10}(2^x-1)$ and ${\log }_{10}(2^x+3)$ be three consecutive term of an A. P., then :

• A. $x=0$
• B. $x=1$
• C. $x={\log }_{2}5$
• D. $x={\log }_{10}2$

Answer 1

Answer: (C)

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

$letA={\log }_{10}2,B={\log }_{10}(2^x-1)andC={\log }_{10}(2^x+3)$

$Now,2B=A+C$

${(2^x-1)}^2=2.(2^x+3)let{2}^x=z$

${(z-1)}^2=2(z+3)$

$z^2-2z+1=2z+6$

$z^2-4z-5=0$

$(z-5)(z+1)=0$

$z=5,-1asz\ne -1,$

$So,z=5$

$2^x=5$

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