Question

The number of positive integers satisfying the equation $x+{\log }_{10}(2^x+1)=x{\log }_{10}5+{\log }_{10}6$ is

• A. $0$
• B. $1$
• C. $2$
• D. infinite

Answer 1

The correct option is B $1$

${\log }_{10}{10}^x+{\log }_{10}(2^x+1)={\log }_{10}{5}^x+{\log }_{10}6$
$\Rightarrow {10}^x(2^x+1)=5^x\cdot 6$
$\Rightarrow 2^x(2^x+1)=6(\because 5^x\ne 0)\Rightarrow (2^x{)}^2+2^x-6=0\Rightarrow 2^x=2\text{or}{2}^x=-3(\text{Not possible})$
$\therefore 2^x=2$
$\Rightarrow x=1$