If $$log_{10} x - log_{10} \sqrt[3]{x} = 6log_{x}10$$ then the value of x is

$$\log_{10} x - \log_{10} \sqrt[3]{x} = 6\log_{x}10$$
Thus, $$\dfrac{\log {x}}{\log {10}}$$ - $$\dfrac{1}{3}*\dfrac{\log {x}}{\log {10}}$$ = $$6*\dfrac{\log{10}}{\log{x}}$$
=> $$\dfrac{2}{3}*\dfrac{\log {x}}{\log {10}}$$ = $$6*\dfrac{\log{10}}{\log{x}}$$
Thus, => $$\dfrac{1}{9}*(\log{x})^2 = (\log{10})^2=1$$
Thus, $$(\log{x})^2 = 9$$  
Thus $$\log x = 3$$ or $$-3$$
Thus, $$ x = 1000$$ or $$\dfrac{1}{1000}$$
From amongst the given options, 1000 is the correct answer. 
Hence, option D is the correct answer.