Log Your Body

Similar solution with @Rahil Sehgal 's.

$\begin{aligned} \log_\frac 14 \log_\frac 13 \log_\frac 12 \sqrt[3]{\frac 1{9x^2-244x+29}} & = 0 \\ \implies \log_\frac 13 \log_\frac 12 \sqrt[3]{\frac 1{9x^2-244x+29}} & = 1 \\ \log_\frac 12 \sqrt[3]{\frac 1{9x^2-244x+29}} & = \frac 13 \\ \sqrt[3]{\frac 1{9x^2-244x+29}} & = \left(\frac 12 \right)^\frac 13 = \sqrt[3]{\frac 12} \\ \implies 9x^2-244x+29 & = 2 \\ 9x^2-244x+27 & = 0 \\ (9x-1)(x-27) & = 0 \\ \implies x & = \frac 19, 27 \end{aligned}$

Therefore the sum of roots are $27\frac 19 \approx \boxed{27.11}$ .