logₐb = log_c b / log_c a（change-of-base formula）. logₐb × log_b c = logₐc（chain rule）.
Example ①
(1) log₄8 (2) log₂3 × log₃5 × log₅4
Answer: (1) log₂8/log₂4=3/2 (2) log₂4=2
(1) log₄8 (2) log₂3 × log₃5 × log₅4
Answer: (1) log₂8/log₂4=3/2 (2) log₂4=2
Example ②
When log₁₀2=0.3010, find log₁₀5.
Answer: log₁₀5=log₁₀(10/2)=1−0.3010=0.6990
When log₁₀2=0.3010, find log₁₀5.
Answer: log₁₀5=log₁₀(10/2)=1−0.3010=0.6990
Key Point
- Change of base: unify the bases as common logarithms for calculation / Chain rule: logₐb×log_bc=logₐc
Practice Problems
- Calculate log₉27.
- Calculate log₂5×log₅8.
- When log₁₀2=0.3010, find log₁₀50.
Answers and Explanations
- Answer: 3/2
- Answer: 3
Explanation: By the chain rule, log₂8=3. - Answer: 1.6990
Explanation: log₁₀50=log₁₀(100/2)=2−0.3010=1.6990.