Question

$\log {}_{10}2,\log {}_{10}(2^x-1)$ and $(2^x+3)$ are three consecutive terms of an A.P for :

• A. more than two real x
• B. no real x
• C. exactly one real x
• D. exactly two real x

Answer 1

The correct option is C exactly one real x

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

$2lo{g}_{10}(2^x-1)=lo{g}_{10}2+lo{g}_{10}(2^x+3)$

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

$\Rightarrow (2^x-1{)}^2=(2\left)\right(2^x+3)$

Let $2^x=a$

$\Rightarrow (a-1{)}^2=2(a+3)$

$\Rightarrow a^2+1-2a=2a+b$

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

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

$\Rightarrow a=5$ or $a=-1$

$2^x=5$ or $2^x=-1$

Exactly one real

x as $2^x\ne -1$ so $2^x=5$

$\Rightarrow xlog2=log5$

$\Rightarrow x=lo{g}_{2}5$.