Question

If $x+{\log }_{10}(1+2^x)=x{\log }_{10}5+{\log }_{10}6$ then $x$ is equal to

• A. $2,-3$
• B. $2$ only
• C. $1$
• D. $3$

Answer 1

The correct option is C $1$

$x+{\log }_{10}(1+2^x)=x{\log }_{10}5+{\log }_{10}6\Rightarrow x(1–{\log }_{10}5)={\log }_{10}6–{\log }_{10}(1+2^x)\Rightarrow x({\log }_{10}10–{\log }_{10}5)={\log }_{10}\left(\frac{6}{1+2^x}\right)\Rightarrow x{\log }_{10}2={\log }_{10}\left(\frac{6}{1+2^x}\right)$

$\Rightarrow 2^x(1+2^x)=6$

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

$\Rightarrow x=1$

Therefore, answer is $1$