Question

The number of roots of the equation $2^x+2^{x-1}+2^{x-2}=7^x+7^{x-1}+7^{x-2}$ is-

Answer 1

$2^x+2^{x-1}+2^{x-2}=7^x+7^{x-1}+7^{x-2}$
$\Rightarrow 2^x(1+\frac{1}{2}+\frac{1}{4})=7^x(1+\frac{1}{7}+\frac{1}{7^2})$
$\Rightarrow 2^x\left(\frac{7}{4}\right)=7^x\left(\frac{57}{49}\right)$
$\Rightarrow (\frac{7}{2}{)}^{x-2}=(\frac{7}{57})$
$\Rightarrow (x-2)log\left(\frac{7}{2}\right)=log\left(\frac{7}{57}\right)$
$\Rightarrow x=2+\frac{log\left(\frac{7}{57}\right)}{log\left(\frac{7}{2}\right)}$
Number of roots of the equation is 1.