WEBVTT

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歡迎觀看對數課程

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寫一寫“對數”這個詞

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這又是一個構造複雜的新單詞

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像“斜邊”

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最好看一下怎麽寫

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我先把筆調出來

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對數

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很容易拼錯

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我之前去MIT

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到那的一個阿卡貝拉團

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名叫Logarhythms

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像音樂旋律

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扯遠了

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對數是什麽？

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最容易理解的解釋

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是

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把它看成

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指數的逆運算

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舉例說明

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2的3次方

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指數部分已經學過

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2的3次方等於8

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重寫一下 這是2不是z

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2的3次方等於8

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由此得出log

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log表示對數

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log以2爲底8的對數等於3

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看看這個運算大家就明白

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怎麽由指數到對數

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也就是說

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log以2爲底8的對數等於幾

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相當於問 2的幾次方等於8？

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因此對數運算的結果

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也可以說對數表達式的結果

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或者說如果要算對數表達式

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應該算出

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2的指數爲幾的時候等於8

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爲3

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再做幾個例子大家就會理解

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log 筆怎麽了

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log以4爲底64的對數等於x

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這跟另一個式子意義相同

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4的x次方等於64

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換一個思路想

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4的幾次方等於64？

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大家知道4的3次方是64

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所以這裡答案是3

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log以4爲底64的對數等於3

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再多做幾個例子

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大家做的題越多

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就理解的更清楚

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對數是個簡單的概念

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但作爲指數的逆運算

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可能不好理解

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因爲指數也不好理解

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log以10爲底1000000的對數

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標上逗號

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等於幾

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這時候該思考的是

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10的幾次方是1000000

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10的幾次方等於

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1後面跟幾個0

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比如10的5次方

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就等於1後面跟5個0

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現在1後面有6個0

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也就是10的6次方

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所以10的6次方等於1000000

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因爲10的6次方等於1000000

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所以log以10爲底1000000的對數等於6

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記住

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6爲指數

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以10爲底得到1000000

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我用了很多種方法講解對數

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我希望

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這其中會有幾種

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能使大家清楚的理解對數

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再做幾個練習

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下面這個會難一點

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log以1/2爲底1/8的對數

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等於x

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大家回想一下

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這個等同於 1/2

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應該是括號

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1/2的x次方等於1/8

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我們知道1/2的3次方

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等於1/8

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所以log以1/2爲底1/8的對數等於3

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再多做幾個題目

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跟前面的不同

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log以x爲底27的對數等於3

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x等於幾？

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跟前面一樣

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這個等同於 x的3次方等於27

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或是x等於27的立方根

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也就是說一個數自乘3次後

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等於27

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大家知道

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這個數是3

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x等於3

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log以3爲底27的對數

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等於3

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再舉一個例子

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我們只用較小的數做練習

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因爲沒有計算器

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只能心算

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log 我先想想

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log以100爲底1的對數是幾？

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比較難

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再來一次

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假設等於問號

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是log以100爲底1的對數

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那麽100的問號次方

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等於1

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這時候應該想

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有沒有一個數作爲指數時

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結果爲1？

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大家記得指數法則中

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也不算是指數法則

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是指數某一節中的內容

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任何數的0次方都等於1

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所以100的0次方等於1

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log以100爲底1的對數

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等於0

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因爲100的0次方等於1

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再問一個問題

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log以2爲底0的對數是幾？

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等於多少？

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也就是

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假設等於x

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2的x次方等於0

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x是多少？

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有沒有2的幾次方

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可以等於0

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沒有 所以它不成立

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不成立或者說無解

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2的無論幾次方

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都不等於0

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類似的情況 log以3爲底

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-1的對數

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假設在實數範圍內

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到目前爲止

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大家接觸的大部分都是實數的運算

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那麽沒有3的幾次方

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等於負數 所以沒有定義

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因此只要對數底爲正數

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要使式子有意義這個數必須大於

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大於或等於 不對

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必須大於0

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不能等於0

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不能爲0或負數

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再做幾個特殊的題目

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還剩一分半鍾

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大家現在可以做

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難度1的對數題了

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我們再做幾個題

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log以8爲底

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稍難一點 1/64的對數是多少

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有趣

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大家知道log以8爲底64的對數等於2