例題 1 Example 1：合併對數 Combine logarithms
化簡 log₂12 + log₂(2/3)。Simplify log₂12 + log₂(2/3).
log₂(12 × 2/3) = log₂88 = 2³
答案： Answer: 3
【對數定律公式：換底公式及解對數方程】Laws of Logarithms: Log Laws, Change of Base and Log Equations
SketchMath 數學團隊 SketchMath Maths Team ｜ 更新 Updated 2026-10-20
loga(MN) = logaM + logaN、loga(M/N) = logaM − logaN、loga(Mᵏ) = k logaM。底數必須相同。換底：logab = log b / log a。loga(MN) = logaM + logaN, loga(M/N) = logaM − logaN, loga(Mᵏ) = k logaM. The base must be the same. To change base: logab = log b / log a.
對數定律 Log Laws
定義Definition
aˣ = N ⟺ x = logaN積Product
loga(MN) = logaM + logaN商Quotient
loga(M/N) = logaM − logaN冪Power
loga(Mᵏ) = k logaM特殊值Special values
loga1 = 0, logaa = 1換底Change of base
logab = logcb / logca對數定律例題詳解 Worked Examples
例題 2 Example 2：對數方程 A logarithmic equation
解 log₃(x + 2) + log₃x = 1。Solve log₃(x + 2) + log₃x = 1.
log₃[x(x + 2)] = 1, sox(x + 2) = 3x² + 2x − 3 = 0, so(x + 3)(x − 1) = 0- 捨去
x = −3（對數的真數必須是正數）Rejectx = −3(the argument of a logarithm must be positive)
答案： Answer: x = 1
例題 3 Example 3：換底 Change of base
求 log₂7，準確至 3 位小數。Evaluate log₂7 correct to 3 decimal places.
log₂7 = log 7 / log 2 = 0.84510 / 0.30103
答案： Answer: 2.807
💡 應試陷阱與秘訣 Exam Traps & Secrets
⚠️ 每個解都要檢驗 Check every solution對數只有在真數為正時才有定義。要把解代回原方程，捨去使真數為 0 或負數的根。A logarithm is only defined for a positive argument. Substitute back and reject roots that make an argument zero or negative.
⚠️ log(M + N) 並不等於 log M + log N log(M + N) is not log M + log N定律把「積」變成「和」，但「和」的對數沒有公式可用。The laws turn products into sums. There is no rule for the logarithm of a sum.
✏️ 自我測試：互動練習 Self-Test: Interactive Practice
溫習方法：點擊以下任何題目，即時在坐標網格載入並核對計算步驟。How to study: click any question to load it on the coordinate grid and check your steps.
(合併 Combine)把 2 log x + 3 log y − log z 寫成單一個對數。Express 2 log x + 3 log y − log z as a single logarithm.
(對數方程 Log equation)解 log₂(x − 1) + log₂(x + 1) = 3。Solve log₂(x − 1) + log₂(x + 1) = 3.
(表示 Express)已知 log 2 = a，log 3 = b，用 a 及 b 表示 log 72。Given log 2 = a and log 3 = b, express log 72 in terms of a and b.
對數定律練習題（DSE 模擬題） Exercises: DSE-Style Practice
SketchMath 題庫內的 DSE 風格練習，選擇題及長題目均附解答。 DSE-style practice from the SketchMath question bank, with solutions for every MC and long question.
指數函數及對數函數 · 中四
47 條選擇題 47 MC · 2 條長題目 2 long questions
常見問題 Frequently Asked Questions
積的對數、商的對數及冪的對數。它們把乘、除及冪分別化為加、減及乘。
The product law, the quotient law and the power law. They turn multiplication, division and powers into addition, subtraction and multiplication.
在計數機上，log 代表底數 10，ln 代表底數 e。
On a calculator, log means base 10 and ln means base e.
兩邊取對數：x = log 20 / log 3 ≈ 2.727。
Take logarithms of both sides: x = log 20 / log 3 ≈ 2.727.