In which of the following choices must \(a\) be less than \(b?\)

[Vincen]

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

[Scott@TargetTestPrep]

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

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