2^24 > 5^10 is quite obvious, you can't calculate it without a computer (Btw, I didn't do so). You can reckon that the number of exponents is so big ( > twice that of RHS) that LHS will be > RHS.
I don't consider that inequality to be obvious at all. If instead you were asked if 2^22 > 5^10 was true, you would be misled if you simply compared the exponents. 2^22 = (2^11)^2 = (2048)^2, whereas 5^10 = (5^5)^2 = (3125)^2, so 5^10 is certainly bigger than 2^22.
I'm not sure there's a convenient way to answer the question in the original post unless you notice that one power of 2, namely 2^10, is very close to a power of 10. Since 2^10 = 1024, then we have:
x > 2^34 = (2^4)(2^10)^3 = 16*(1024)^3 > 16*(10^3)^3 = 16*(10^9) > 10*10^9 = 10^10
so x > 10^10, and Statement 1 is sufficient. It's such a specialized 'trick' that we need here that I don't consider the question particularly valuable to study - you are extremely unlikely to need to do anything similar on the actual GMAT.
Statement 2 is clearly sufficient without doing any work: we know the value of x, so we can compare it to any number at all.
The answer is D.
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