My problems #10

Algebra Level 3

The sum of the infinite series

$\dfrac{\text{log}2}{3}+\dfrac{\text{log}4}{9}+\dfrac{\text{log}8}{27}+\ldots$

can be expressed as $\displaystyle\ \dfrac{a\times{\text{log}\ b}}{c}.$ Find $a+b+c$