Fix the mistake: Logarithmic Functions (Definition, Properties, and Logarithmic Equations)

Definition and Basic Properties of Logarithms

a^p=M ⟺ logₐM=p. Logarithm calculations: products→sums, quotients→differences, powers→coefficients.

Basic Properties

  • logₐ1=0 / logₐa=1
  • logₐ(MN)=logₐM+logₐN
  • logₐ(M/N)=logₐM−logₐN
  • logₐMᵏ=k logₐM
Example ①
(1) log₂32　(2) log₃(1/27)　(3) log₅1
Answer: (1) 5　(2) −3　(3) 0
Example ②
When log₂3=a, express log₂12 in terms of a.
Answer: log₂(4×3)=2+a
Key Point
  • “What power of a gives M?” is logₐM / a^(logₐM)=M

Practice Problems

  1. Find (1) log₂(1/4)　(2) log₈2　(3) log₇49.
  2. Express log₁₀18 in terms of log₁₀2=a and log₁₀3=b.
  3. Calculate log₂6+log₂(2/3).

Answers and Explanations

  1. Answer: (1) −2　(2) 1/3　(3) 2
  2. Answer: a+2b
  3. Answer: 2
    Explanation: log₂(6×2/3)=log₂4=2.

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