jmc

We can write $$\begin{array}{rl}& 17{\log }_{30}x-3{\log }_{x}5+20{\log }_{x}15-3{\log }_{x}6+20{\log }_{x}2\\ & =17{\log }_{30}x-{\log }_x{5}^3+{\log }_{x}15^{20}-{\log }_x{6}^3+{\log }_x{2}^{20}\\ & =17{\log }_{30}x+{\log }_x\frac{15^{20}\cdot 2^{20}}{5^3\cdot 6^3}\\ & =17{\log }_{30}x+{\log }_x(2^{17}\cdot 3^{17}\cdot 5^{17})\\ & =17{\log }_{30}x+17{\log }_{x}30\\ & =17\left({\log }_{30}x+\frac{1}{{\log }_{30}x}\right).\end{array}$$By AM-GM, $${\log }_{30}x+\frac{1}{{\log }_{30}x}\ge 2,$$so $17\left({\log }_{30}x+\frac{1}{{\log }_{30}x}\right)\ge 34.$ Equality occurs when $x=30,$ so the minimum value is $\boxed{34}.$