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第 13 週 · 指數與對數 · 第 6 節

對數的應用(地震 · 聲音 · 酸鹼)+ 本週關卡

對數不只是符號 —— 地震、聲音、酸鹼這些「跨好幾個數量級」的量都用它來度量,因為它把 ×10 的暴漲壓成 +1 的台階。本節講透三大應用,再以本週 BOSS 總關卡收官。

▶ 第一關: 里氏地震震級(每級 ×10)
Richter scale: each +1 level = amplitude x10R1 x10R2 x100R3 x1000R4 x10000R5 x100000R6 x1000000+1 levellabel x10 each step
level difference -> amplitude ratiodiff 1x10diff 2x100diff 3x1000diff nx10^n
▶ 第二關: 分貝聲音 & pH 酸鹼
Decibels: each +10 dB = intensity x100 dBx110 dBx1020 dBx10030 dBx1000+10 dB = x10+10 dB = x10
pH: each -1 in pH = acidity x10pH2pH4pH6pH8more acid (smaller pH)more alkalinepH 3 vs pH 5: down 2 = x100 acidity
▶ 第三關: 對數刻度與應用公式
Linear scale vs Log scale (same numbers)LINEAR: gaps blow up, 1-10 squashed1101001000LOG: evenly spaced, each step x101101001000x10x10x10log turns each x10 jump into one +1 step
One formula behind all threeEarthquakeR = log(A/A0)SoundL = 10 log(I/I0)AciditypH = -log[H+]all are log( quantity / reference )
⚔ BOSS: W13 本週總關卡
W13 map: Exponents and LogarithmsD1 rational exponents a^(m/n)D2 what is a logarithmD3 log laws and change of baseD4 exp and log graphsD5 solving exp and log eqnsD6 applications quake sound pHBOSS below reviews five core skills
BOSS: five W13 skills in three questionsQ13 rational exponent8^(4/3) = 2^4 = 16Q14 log value + lawlog20+log5 = log100 = 2Q15 solve equation + application4 = log(A/A0) -> A/A0 = 10^4 = 10000exponents | logs | laws | equations | appsclear all three to finish week 13