Trig Identities and Logs, Solve x within the given interval

Note: I wasn't really sure which board to post this on since this involves both trig identities and logs, but I figured that since Trigonometry generally comes after Algebra that I'd post it in this board. If I put this topic in the wrong board, I apologize for my confusion.

Solve for x in the interval [0, 2$\displaystyle \pi$]:

$\displaystyle \log_{2}\cot x - 2\log_{4}\csc2x = \log_{2}\cos x$

Now, so far I've tried turning the $\displaystyle 2log_{4}$ into $\displaystyle 4\log_{2}$ and then getting stuck with not knowing how to break down $\displaystyle csc 2x$ so I can use log identities (subtracting logs with same bases becomes division of factors, for reference). I've also tried breaking it down to the following, but I hit a dead end because it doesn't really seem to lead anywhere that I can tell:

$\displaystyle \log_{2}csc x = \log_{4}csc4x^2$

The answer is supposed to be $\displaystyle x = \frac{\pi}{3} , \frac{5\pi}{3}$

So, any tips on how to progress, or which way to progress, would be appreciated. Thanks. (Nod)