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第 13 週 · 指數與對數 · 第 4 節

指數函數與對數函數的圖像

把指數 y = aˣ 和對數 y = logₐx 畫成圖,性質一眼看清 —— 過哪點、升還是降、定義域值域、為何是「鏡像兄弟」,配滿圖例從零講透。

▶ 第一關: 指數函數 y = aˣ 的圖像
xy(0, 1)(1, 2)(2, 4)(3, 8)y = 2ˣ (a > 1, rises fast)
(0,1)a > 1 : increasing ↗(0,1)0 < a < 1 : decreasing ↘
range: y > 0(curve stays ABOVE x-axis)x-axis = horizontal asymptotenever touches 0
▶ 第二關: 對數函數 y = logₐx 的圖像
xy(1, 0)(2, 1)(4, 2)(8, 3)y = log₂ x (rises slowly)
x ≤ 0FORBIDDENdomain: x > 0 onlyy-axis = vertical asymptote
▶ 第三關: 鏡像關係 + 讀圖比較
xyy = xy = 2ˣy = log₂ x(0,1)(1,0)(1,2)(2,1)(3,8)(8,3)each point flips across y = x
y = 2ˣ pointy = log₂ x point(0, 1)(3, 8)swap →swap →(1, 0)(8, 3)
EXPONENTIAL y = aˣpasses (0, 1)hugs x-axisdomain: all reals / range: y > 0LOG y = logₐ xpasses (1, 0)hugs y-axisdomain: x > 0 / range: all reals
⚔ BOSS: 綜合判斷圖像性質 / 給點求底
given: y = logₐ x passes (9, 2)logₐ 9 = 2a² = 9a = 3check: log₃ 9 = 2 because 3² = 9 ✓