In the interval& (0, 2π) , sum of all the roots of the equation (π...

\pi { log}_{3} \left(\dfrac{1}{\text{x}}\right) = \text{k} \pi \text{,} \text{k} \in \text{I}$<br /><br />$\log_{3} \left(\dfrac{1}{\text{x}}\right) = \text{k} ⇒ \text{x} = {3}^{- \text{k}}$<br /><br />Possible values of$k$are$– 1, 0, 1, 2, 3, . . . . . . .

\text{S} =\left(3 + 1\right)$+ \left(\dfrac{1}{3} + \dfrac{1}{{3}^{2}} + \dfrac{1}{{3}^{3}} + \text{.....} \infty\right)$

~ $= 4 + \dfrac{\left(1 / 3\right)}{1 - \left(1 / 3\right)} = 4 + \dfrac{1}{2} = \dfrac{9}{2}$