Set the bases equal and compare the exponents: aˢ=aᵗ ⟺ s=t（a>0, a≠1）
Example ①
(1) 2^x=8 (2) 9^x=27 (3) (1/4)^x=32
Answer: (1) x=3 (2) x=3/2 (3) x=−5/2
(1) 2^x=8 (2) 9^x=27 (3) (1/4)^x=32
Answer: (1) x=3 (2) x=3/2 (3) x=−5/2
Example ② (Substitution)
Solve 4^x−3×2^x−4=0.
Answer: Let t=2^x（t>0）. t²−3t−4=0→(t−4)(t+1)=0→t=4→x=2
Solve 4^x−3×2^x−4=0.
Answer: Let t=2^x（t>0）. t²−3t−4=0→(t−4)(t+1)=0→t=4→x=2
Key Point
- Set the bases equal / Substitute t=2^x and narrow the range using t>0
Practice Problems
- Solve (1) 3^(2x+1)=27 (2) 4^x=8^(x−1).
- Solve 9^x−4×3^x+3=0.
- Find the minimum value of f(x)=4^x−2^(x+1)+3.
Answers and Explanations
- Answer: (1) x=1 (2) x=3
- Answer: x=0, 1
Explanation: Let t=3^x. (t−1)(t−3)=0→t=1→x=0、t=3→x=1. - Answer: Minimum value 2（t=2^x、f=(t−1)²+2、the minimum occurs at t=1, that is, x=0）