If , which of the following must be true?
I.
II.
III.
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Wait, so if 3^x > 10, then x has to be greater than 2? Because 3^2 = 9, which is less than 10. So x must be at least slightly more than 2. But then why isn't III also true? Like if x > 2, doesn't that mean x > 4? No, that's not right. Let me think...
You're confusing necessary vs sufficient. If x > 4, then definitely x > 2, but the reverse isn't true. Since 3^x > 10 only forces x > ~2.1, III (x > 4) doesn't have to be true. For example, x = 3 works but is not > 4.
Exactly. And II says x > 3, but x could be 2.5, which still gives 3^2.5 > 10. So II isn't necessary either.
This one got me. I initially thought all three because I was like 'if it's bigger than 10, x must be big'. But then I plugged in x=2.1 and 3^2.1 ≈ 10.05, so it works. So only I must be true. Tricky!
Nice question. It really tests whether you understand 'must be true' vs 'could be true'. I almost picked E but then remembered that x can be between 2 and 3.
What it tests
Your grasp of exponent rules — multiplying and dividing powers, power-of-a-power, and negative/zero exponents.
Common trap
Adding exponents when multiplying bases that have the same exponent, or confusing (x^a)^b with x^(a·b).