a>1：aˢ>aᵗ⟺s>t（increasing）. 0<a<1：aˢ>aᵗ⟺s<t（decreasing; reverse the inequality）
Example ①
(1) 2^x>16 (2) (1/3)^x≤9
Answer: (1) x>4 (2) x≥−2
(1) 2^x>16 (2) (1/3)^x≤9
Answer: (1) x>4 (2) x≥−2
Example ② (Substitution)
Solve 4^x−5×2^x+4≤0.
Answer: t=2^x. (t−1)(t−4)≤0→1≤t≤4→0≤x≤2
Solve 4^x−5×2^x+4≤0.
Answer: t=2^x. (t−1)(t−4)≤0→1≤t≤4→0≤x≤2
Key Point
- Determine the direction of the inequality from the size of the base / After substitution, convert the range of t into the range of x
Practice Problems
- Solve (1) 3^x<81 (2) (1/2)^(x+1)≥4.
- Solve 9^x−10×3^x+9≤0.
- Solve 4^x−2^(x+2)>0.
Answers and Explanations
- Answer: (1) x<4 (2) x≤−3
- Answer: 0≤x≤2
Explanation: Let t=3^x. (t−1)(t−9)≤0→1≤t≤9=3²→0≤x≤2. - Answer: x>2
Explanation: (2^x)²−4×2^x>0→t(t−4)>0→t>4=2²→x>2.