This is very definitely not a GMAT question - logarithms and trig functions are not part of the GMAT. So GMAT test takers should definitely ignore it.

I'm confused looking at the problem, for two reasons - I don't know why the '2' in front of the first log is written as a superscript. I assume we're multiplying the first log by 2? And I don't know why answers B and C are exactly identical.

Anyway, if I've interpreted the question correctly, I was able to to answer it very quickly just by looking at the answer choices, so I didn't consider other approaches. Since you've posted the question, I'll assume you know about those things - if not, nothing below will make much sense!

Since you cannot take a log of a negative or of zero, the first log, log(1 - 2sin(x), will be undefined when 2sin(x) > 1/2. So it will be undefined when π/6 < x < 5π/6. So that rules out answer D. The other answer choices all overlap substantially. They are only different in that some include values between π and 7π/6. So if we could work out whether the inequality is true for values between π and 7π/6, we could pick the right answer. And if we plug in a value very close to, but just less than, 7π/6, then 2sin(x) will be very close to -1, and log(1 + 2sinx) will be extremely close to log(0), so becomes a very negative number. The other log will just be a small positive number, so when we add the two logs, we'll definitely get a negative answer. So the inequality will be true for values of x just below 7π/6, and the answer should be B or C - I can't tell you which because the two answer choices are exactly the same!

I'm sure there are algebraic things you could do here as well. What are you reviewing these questions for?