Question

If $log2,log\left(2^x-1\right)$and $log\left(2^x+3\right)$ are in $A.P$., then the value of $x$ is

• A. $5/3$
• B. $lo{g}_{2}5$
• C. $lo{g}_{3}5$
• D. $lo{g}_{5}3$

Answer 1

Answer: (B)

$2log\left(2^x-1\right)=log2+log\left(2^x+3\right)$
$\Rightarrow {\left(2^x-1\right)}^2=2\cdot \left(2^x+3\right)$
$\Rightarrow {\left(2^x\right)}^2-4\cdot 2^x-5=0$
$\Rightarrow \left(2^x-5\right)\left(2^x+1\right)=0$
$\Rightarrow x=lo{g}_{2}5$, as $2^x+1\ne 0$