Let S be the set of all real roots of the equation, $3^x\left(3^x-1\right)+2=\left|3^x-1\right|+\left|3^x-2\right|$. Then S:

Let $3^x=t$, then t > 0 and

$\begin{array}{l}t(t-1)+2=|t-1|+|t-2|\\ t^2-t+2=|t-1|+|t-2|\end{array}$

On + ve side of x – axis, there is only one point of intersection   

$\therefore t=a\Rightarrow x={\log }_{3}a$