例題 1 Example 1：求底數 Find the base
y = logax 的圖像經過 (8, 3)，求 a。The graph of y = logax passes through (8, 3). Find a.
loga8 = 3即a³ = 8loga8 = 3meansa³ = 8
答案： Answer: a = 2
【對數函數圖像：y = log_a x】Logarithmic Functions and Graphs: y = log_a x
SketchMath 數學團隊 SketchMath Maths Team ｜ 更新 Updated 2026-10-20
y = logax 的定義域是 x > 0，圖像經過 (1, 0) 及 (a, 1)，y 軸是它的垂直漸近線。它是 y = aˣ 沿直線 y = x 的反射，所以 a > 1 時上升，0 < a < 1 時下降。The graph of y = logax has domain x > 0, passes through (1, 0) and (a, 1), and has the y-axis as its vertical asymptote. It is the reflection of y = aˣ in the line y = x, so it rises when a > 1 and falls when 0 < a < 1.
對數函數圖像重點 Logarithmic Graph Facts
定義域及值域Domain and range
x > 0, all real y圖像上的點Points on the graph
(1, 0) and (a, 1)漸近線Asymptote
x = 0 (the y-axis)反函數Inverse function
y = aˣ ⟷ y = logax平移後的圖像Shifted graph
y = loga(x − h) 的漸近線是 x = h，定義域是 x > hy = loga(x − h) has asymptote x = h and domain x > h平移圖像：定義域及漸近線 Shifting the Graph: Domain and Asymptote
y = loga(x − h) 的圖像是 y = logax 向右移 h 個單位。它的垂直漸近線是 x = h，定義域是 x > h。
The graph of y = loga(x − h) is y = logax moved h units to the right. Its vertical asymptote is x = h and its domain is x > h.
y = aˣ 的反函數 The Inverse of y = aˣ
對數是指數的反運算：若 (t, 2ᵗ) 在 y = 2ˣ 上，則 (2ᵗ, t) 在 y = log₂x 上。兩個圖像沿直線 y = x 互為鏡像。
The logarithm undoes the exponential: if (t, 2ᵗ) is on y = 2ˣ, then (2ᵗ, t) is on y = log₂x. The two graphs are mirror images in the line y = x.
對數函數圖像例題詳解 Worked Examples
例題 2 Example 2：定義域及漸近線 Domain and asymptote
寫出 y = log₂(x − 3) 的定義域及漸近線。State the domain and asymptote of y = log₂(x − 3).
- 真數必須是正數：
x − 3 > 0The argument must be positive:x − 3 > 0 y = log₂x的圖像向右移了 3 個單位The graph ofy = log₂xhas moved 3 units to the right
答案： Answer: 定義域 x > 3；漸近線 x = 3Domain x > 3; asymptote x = 3
例題 3 Example 3：反函數 Inverse function
求 y = 3ˣ 的反函數。Find the inverse of y = 3ˣ.
- 交換 x 與 y：
x = 3ʸSwap x and y:x = 3ʸ - 把 y 寫成對數Write y as a logarithm
答案： Answer: y = log₃x
💡 應試陷阱與秘訣 Exam Traps & Secrets
⚠️ 先看定義域 Domain first對數只對正數有定義，所以圖像永不碰到 y 軸，亦不會出現在 y 軸左方。A logarithm exists only for positive arguments, so the graph never reaches the y-axis or goes to its left.
⚠️ 混淆兩條漸近線 Mixing up the two asymptotes指數函數的漸近線是水平的（
y = 0）；對數函數的漸近線是垂直的（x = 0）。The exponential graph has a horizontal asymptote (y = 0); the logarithmic graph has a vertical one (x = 0).✏️ 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.
(畫圖 Sketch)畫 y = log₂x 及 y = log₂(x + 1) − 2，並寫出漸近線。Sketch y = log₂x and y = log₂(x + 1) − 2 and state the asymptotes.
(求底數 Find the base)y = log_a(x + 2) 的圖像經過 (6, 2)，求 a。The graph of y = log_a(x + 2) passes through (6, 2). Find a.
(反函數 Inverse)求 y = 2·5^x + 1 的反函數。Find the inverse function of y = 2·5^x + 1.
對數函數圖像練習題（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
所有正實數：y = log_a x 的定義域是 x > 0。
All positive real numbers: x > 0 for y = log_a x.
圖像緩慢上升，經過 (1, 0)，並從右方無限接近 y 軸但不接觸。
It rises slowly, passes through (1, 0) and approaches the y-axis from the right without touching it.
兩者沿直線 y = x 互為鏡像。
The two are mirror images in the line y = x.