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第 13 週 · 指數與對數 · 第 3 節

對數定律與換底(log 把運算降級)

對數最厲害的地方 —— 它把「難的運算降一級」:乘變加、除變減、冪變乘。三條定律 + 一條換底公式,配滿圖例一次講透,讓你用 log 把複雜運算化簡求值。

▶ 第一關: 乘法 → 加法、除法 → 減法
logₐ(M × N)multiply inside=logₐM + logₐNadd outside (× drops to +)LAW 1 : multiplication becomes addition
logₐ(M / N)divide inside=logₐM − logₐNsubtract outside ( / drops to − )LAW 2 : division becomes subtraction
log₂8 + log₂2log₂(8×2)log₂16 = 4two logs added → one log of the product → read off (2⁴ = 16)
WRONG — cannot split a +logₐ(M+N) ≠ logₐM+logₐNlog₂(8+8)=4 but 3+3=6OK — only × and /logₐ(MN) = logₐM+logₐNinside must be × or /the laws only break apart multiplication / division
▶ 第二關: 冪 → 乘(降級階梯)
logₐ(Mᵏ)power inside=k × logₐMpower drops to front as ×kLAW 3 : a power becomes a multiplier
log lowers every operation by ONE levelpower Mᵏmultiply M × Nadd M + N→ ×k→ +easiestdown 1down 1
▶ 第三關: 合併化簡(反方向)
log₆4 + log₆9log₆(4×9)log₆36 = 2+ outside → × insidecombine to one log, then read (6² = 36)
log₂6 + log₂4 − log₂3log₂(6×4÷3)=log₂8= 3+ → × , − → ÷ (one bracket, then evaluate)
⚔ BOSS: 定律綜合 + 換底公式
logᵦN=logₐNlogₐb← original number← the old base bpick any new base a — top and bottom share that SAME base
changing base keeps the VALUE — only the form changeslog₄64=log₂64 / log₂4both equal 3= 6 / 2 = 3