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第 13 週 · 指數與對數 · 第 2 節

對數是什麼（指數的逆運算）

上節把指數玩熟了；這節學它的「逆運算」—— 對數。已知底和結果，反求指數。logₐb = c 到底在問什麼？用滿滿圖例把定義講透。

▶ 第一關： 對數的定義（log 在問什麼）
logₐb = c⟺aᶜ = blogarithm formexponential formtwo ways to write the SAME relationship
logab= ?base (the bottom number)argument (must be > 0)
log₂ 8 = ?2 to what power = 8 ?→2³ = 8answer = 3
▶ 第二關： 逆運算、互換與特殊值
EXP machine (base 2)38LOG machine (base 2)83they undo each other: exponent in / out
2³ = 8exponential⟺log₂8 = 3logarithmbase 2 stays the base; the 3 is the exponentbase ↓ , exponent = the log value , result = argument
logₐ 1 = 0because a⁰ = 1logₐ a = 1because a¹ = a
▶ 第三關： 求值練習（含常用對數）
log₃ 81→81 = 3⁴→answer 4write the argument as a power of the base
log b = log₁₀ bcommon log, base 10ln b = logₑ bnatural log, base e ≈ 2.718
⚔ BOSS： 對數綜合求值
fractional argumentlog₂(1/8) = −32⁻³ = 1/8fractional answerlog₉3 = 1/29^(1/2) = 3