Equation: logₐM=logₐN⟺M=N（compare the arguments）. Inequality: unchanged when a>1; reversed when 0<a<1.
Example ①
Solve log₂(x+3)=3.
Answer: x+3=8→x=5（check: 8>0✓）
Solve log₂(x+3)=3.
Answer: x+3=8→x=5（check: 8>0✓）
Example ②
Solve log₂x+log₂(x−2)=3.
Answer: x(x−2)=8→x²−2x−8=0→(x−4)(x+2)=0. Since x>2, x=4
Solve log₂x+log₂(x−2)=3.
Answer: x(x−2)=8→x²−2x−8=0→(x−4)(x+2)=0. Since x>2, x=4
Example ③
Solve log₃(x+1)>2.
Answer: Since the base 3>1, the inequality is unchanged: x+1>9→x>8（together with the argument condition x>−1, x>8）
Solve log₃(x+1)>2.
Answer: Since the base 3>1, the inequality is unchanged: x+1>9→x>8（together with the argument condition x>−1, x>8）
Key Point
- Always check the argument condition / For an inequality, reverse the inequality when the base is less than 1
Practice Problems
- Solve log₃(2x−1)=2.
- Solve log₂(x+1)+log₂(x−1)=3.
- Solve log_(1/2)(x+3)≤−1.
Answers and Explanations
- Answer: x=5
- Answer: x=3
Explanation: (x+1)(x−1)=8→x²=9→x=±3. Since x>1, x=3. - Answer: x≥−1
Explanation: Since the base 1/2<1, reverse the inequality: x+3≥(1/2)^(−1)=2→x≥−1. Together with the argument condition x>−3, x≥−1.