In which of the following choices must \(a\) be less than \(b?\)
(A) \(4^{-2b}=4^{-a}\)
(B) \(-4^b<4^a\)
(C) \(4^{-b} < 4^{-a}\)
(D) \(4^{-b}>4^{-a}\)
(E) \(4^{-b}=4^{-2a}\)
Answer: C
Source: Princeton Review
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In which of the following choices must \(a\) be less than \(b?\)
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Solution:
Recall that if x > 1, then x^m ≥ x^n if and only if m ≥ n (and the ≥ sign can be replaced by the >, < or ≤ sign). Therefore, let’s check choice C first:
Since 4^(-b) < 4^(-a), we have:
-b < -a
Dividing both sides by -1 and switching the inequality sign, we have:
b > a (which is the same as a < b)
Answer: C
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