對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件

對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件提示：3－3提示：不存在．提示：利用對(duì)數(shù)求解．提示：3－3提示：不存在．提示：利用對(duì)數(shù)求解．

1．對(duì)數(shù)的概念

(1)定義：如果ax＝N(a>0，且a≠1)，那么數(shù)

叫做以

為底

的對(duì)數(shù)，記作

．其中，

叫做對(duì)數(shù)的底數(shù)，

叫做真數(shù)．xaNx＝logaNaN1．對(duì)數(shù)的概念xaNx＝logaNaN

(2)常用對(duì)數(shù)與自然對(duì)數(shù)：通常將以10為底的對(duì)數(shù)叫做常用對(duì)數(shù)，并把log10N記作

；以無(wú)理數(shù)e＝2.71828…為底數(shù)的對(duì)數(shù)稱(chēng)為自然對(duì)數(shù)，并且把logeN記為

．lgNlnN(2)常用對(duì)數(shù)與自然對(duì)數(shù)：lgNlnN

2．對(duì)數(shù)與指數(shù)的關(guān)系當(dāng)a＞0，且a≠1時(shí)，ax＝N?

．前者叫指數(shù)式，后者叫對(duì)數(shù)式．x＝logaN2．對(duì)數(shù)與指數(shù)的關(guān)系x＝logaN3．對(duì)數(shù)的性質(zhì)負(fù)數(shù)零00113．對(duì)數(shù)的性質(zhì)負(fù)數(shù)零0011對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件問(wèn)題1：我們知道am＋n＝am·an，那么loga（

M·N）

＝logaM·logaN正確嗎？舉例說(shuō)明．提示：不正確，例如log24＝log22×2＝log22·log22＝1×1＝1，而log24＝2.問(wèn)題1：我們知道am＋n＝am·an，那么l問(wèn)題2：你能推出loga（M·N）(M>0，N>0)的表達(dá)式嗎？提示：能．令am＝M，an＝N，∴MN＝am＋n.由對(duì)數(shù)的定義知logaM＝m，logaN＝n，logaMN＝m＋n，∴l(xiāng)ogaMN＝logaM＋logaN.問(wèn)題2：你能推出loga（M·N）(M>0，N>0)的表logaM＋logaNlogaM－logaNnlogaMlogaM＋logaNlogaM－logaNnlogaM2．換底公式若c>0且c≠1，則logab＝

．2．換底公式

1．根據(jù)對(duì)數(shù)的定義，對(duì)數(shù)logaN就是方程ax＝N的解的一個(gè)記號(hào)．因此，alogaN＝N，此式稱(chēng)為對(duì)數(shù)恒等式(其中a>0，且a≠1，N>0)．

2．在應(yīng)用對(duì)數(shù)的運(yùn)算性質(zhì)時(shí)應(yīng)注意保證每個(gè)對(duì)數(shù)式都有意義，應(yīng)避免出現(xiàn)類(lèi)似log2(－7)2＝2log2(－7)的錯(cuò)誤．同時(shí)注意對(duì)數(shù)性質(zhì)在解題中的逆用．1．根據(jù)對(duì)數(shù)的定義，對(duì)數(shù)logaN就是方程ax＝N的解的

3．對(duì)數(shù)的運(yùn)算性質(zhì)概括為：積的對(duì)數(shù)等于對(duì)數(shù)的和；商的對(duì)數(shù)等于對(duì)數(shù)的差，冪的對(duì)數(shù)等于冪的指數(shù)乘以?xún)绲讛?shù)的對(duì)數(shù)．3．對(duì)數(shù)的運(yùn)算性質(zhì)概括為：積的對(duì)數(shù)等于對(duì)數(shù)的和；商的對(duì)數(shù)對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件

[一點(diǎn)通]

1．在利用ax＝N?x＝logaN（a＞0且a≠1）進(jìn)行互化時(shí)，關(guān)鍵是弄清各個(gè)字母所在的位置．

2．對(duì)數(shù)式與指數(shù)式的關(guān)系如圖：[一點(diǎn)通]對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件

[一點(diǎn)通]

1．對(duì)于底數(shù)相同的對(duì)數(shù)式的化簡(jiǎn)或求值，常用的方法是

(1)“收”，將同底的對(duì)數(shù)的和(差)收成積(商)的對(duì)數(shù)；

(2)“拆”，將積(商)的對(duì)數(shù)拆成對(duì)數(shù)的和(差)．對(duì)數(shù)的化簡(jiǎn)或求值一般是正用或逆用公式，對(duì)真數(shù)進(jìn)行處理.選哪種策略化簡(jiǎn)，取決于問(wèn)題的實(shí)際情況，一般本著便于真數(shù)化簡(jiǎn)的原則進(jìn)行．[一點(diǎn)通]

2．loga1＝0，logaa＝1(a>0，且a≠1)在計(jì)算對(duì)數(shù)值時(shí)經(jīng)常用到．2．loga1＝0，logaa＝1(a>0，且a≠1)在對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件

[一點(diǎn)通]

利用對(duì)數(shù)的換底公式能夠?qū)⒉煌椎膶?duì)數(shù)化為常用對(duì)數(shù)或自然對(duì)數(shù)或同底的對(duì)數(shù)，即可用對(duì)數(shù)的運(yùn)算性質(zhì)來(lái)解決對(duì)數(shù)求值問(wèn)題，同時(shí)要注意換底公式的逆用．[一點(diǎn)通]利用對(duì)數(shù)的換底公式能夠?qū)⒉煌椎膶?duì)數(shù)化為常用對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件

[一點(diǎn)通]

對(duì)數(shù)式的證明和對(duì)數(shù)式的化簡(jiǎn)的基本思路是一致的，就是根據(jù)對(duì)數(shù)的運(yùn)算性質(zhì)和換底公式對(duì)對(duì)數(shù)式化簡(jiǎn)．[一點(diǎn)通]對(duì)數(shù)式的證明和對(duì)數(shù)式的化簡(jiǎn)的基本思路是一致的

1．換底公式可完成不同底數(shù)的對(duì)數(shù)式之間的轉(zhuǎn)化.該公式既可正用，又可逆用.使用時(shí)，關(guān)鍵是選擇底數(shù).換底的目的是利用對(duì)數(shù)的運(yùn)算性質(zhì)對(duì)對(duì)數(shù)式進(jìn)行化簡(jiǎn)．

2．應(yīng)用對(duì)數(shù)的運(yùn)算性質(zhì)應(yīng)注意的問(wèn)題

(1)在各對(duì)數(shù)有意義的前提下才能應(yīng)用運(yùn)算性質(zhì)．

(2)根據(jù)不同的問(wèn)題選擇公式的正用或逆用．1．換底公式可完成不同底數(shù)的對(duì)數(shù)式之間的轉(zhuǎn)化.該公式既可對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件提示：3－3提示：不存在．提示：利用對(duì)數(shù)求解．提示：3－3提示：不存在．提示：利用對(duì)數(shù)求解．

1．對(duì)數(shù)的概念

(1)定義：如果ax＝N(a>0，且a≠1)，那么數(shù)

叫做以

為底

的對(duì)數(shù)，記作

．其中，

叫做對(duì)數(shù)的底數(shù)，

叫做真數(shù)．xaNx＝logaNaN1．對(duì)數(shù)的概念xaNx＝logaNaN

(2)常用對(duì)數(shù)與自然對(duì)數(shù)：通常將以10為底的對(duì)數(shù)叫做常用對(duì)數(shù)，并把log10N記作

；以無(wú)理數(shù)e＝2.71828…為底數(shù)的對(duì)數(shù)稱(chēng)為自然對(duì)數(shù)，并且把logeN記為

．lgNlnN(2)常用對(duì)數(shù)與自然對(duì)數(shù)：lgNlnN

2．對(duì)數(shù)與指數(shù)的關(guān)系當(dāng)a＞0，且a≠1時(shí)，ax＝N?

．前者叫指數(shù)式，后者叫對(duì)數(shù)式．x＝logaN2．對(duì)數(shù)與指數(shù)的關(guān)系x＝logaN3．對(duì)數(shù)的性質(zhì)負(fù)數(shù)零00113．對(duì)數(shù)的性質(zhì)負(fù)數(shù)零0011對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件問(wèn)題1：我們知道am＋n＝am·an，那么loga（

M·N）

＝logaM·logaN正確嗎？舉例說(shuō)明．提示：不正確，例如log24＝log22×2＝log22·log22＝1×1＝1，而log24＝2.問(wèn)題1：我們知道am＋n＝am·an，那么l問(wèn)題2：你能推出loga（M·N）(M>0，N>0)的表達(dá)式嗎？提示：能．令am＝M，an＝N，∴MN＝am＋n.由對(duì)數(shù)的定義知logaM＝m，logaN＝n，logaMN＝m＋n，∴l(xiāng)ogaMN＝logaM＋logaN.問(wèn)題2：你能推出loga（M·N）(M>0，N>0)的表logaM＋logaNlogaM－logaNnlogaMlogaM＋logaNlogaM－logaNnlogaM2．換底公式若c>0且c≠1，則logab＝

．2．換底公式

1．根據(jù)對(duì)數(shù)的定義，對(duì)數(shù)logaN就是方程ax＝N的解的一個(gè)記號(hào)．因此，alogaN＝N，此式稱(chēng)為對(duì)數(shù)恒等式(其中a>0，且a≠1，N>0)．

2．在應(yīng)用對(duì)數(shù)的運(yùn)算性質(zhì)時(shí)應(yīng)注意保證每個(gè)對(duì)數(shù)式都有意義，應(yīng)避免出現(xiàn)類(lèi)似log2(－7)2＝2log2(－7)的錯(cuò)誤．同時(shí)注意對(duì)數(shù)性質(zhì)在解題中的逆用．1．根據(jù)對(duì)數(shù)的定義，對(duì)數(shù)logaN就是方程ax＝N的解的

3．對(duì)數(shù)的運(yùn)算性質(zhì)概括為：積的對(duì)數(shù)等于對(duì)數(shù)的和；商的對(duì)數(shù)等于對(duì)數(shù)的差，冪的對(duì)數(shù)等于冪的指數(shù)乘以?xún)绲讛?shù)的對(duì)數(shù)．3．對(duì)數(shù)的運(yùn)算性質(zhì)概括為：積的對(duì)數(shù)等于對(duì)數(shù)的和；商的對(duì)數(shù)對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件

[一點(diǎn)通]

1．在利用ax＝N?x＝logaN（a＞0且a≠1）進(jìn)行互化時(shí)，關(guān)鍵是弄清各個(gè)字母所在的位置．

2．對(duì)數(shù)式與指數(shù)式的關(guān)系如圖：[一點(diǎn)通]對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件

[一點(diǎn)通]

1．對(duì)于底數(shù)相同的對(duì)數(shù)式的化簡(jiǎn)或求值，常用的方法是

(1)“收”，將同底的對(duì)數(shù)的和(差)收成積(商)的對(duì)數(shù)；

(2)“拆”，將積(商)的對(duì)數(shù)拆成對(duì)數(shù)的和(差)．對(duì)數(shù)的化簡(jiǎn)或求值一般是正用或逆用公式，對(duì)真數(shù)進(jìn)行處理.選哪種策略化簡(jiǎn)，取決于問(wèn)題的實(shí)際情況，一般本著便于真數(shù)化簡(jiǎn)的原則進(jìn)行．[一點(diǎn)通]

2．loga1＝0，logaa＝1(a>0，且a≠1)在計(jì)算對(duì)數(shù)值時(shí)經(jīng)常用到．2．loga1＝0，logaa＝1(a>0，且a≠1)在對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件

[一點(diǎn)通]

利用對(duì)數(shù)的換底公式能夠?qū)⒉煌椎膶?duì)數(shù)化為常用對(duì)數(shù)或自然對(duì)數(shù)或同底的對(duì)數(shù)，即可用對(duì)數(shù)的運(yùn)算性質(zhì)來(lái)解決對(duì)數(shù)求值問(wèn)題，同時(shí)要注意換底公式的逆用．[一點(diǎn)通]利用對(duì)數(shù)的換底公式能夠?qū)⒉煌椎膶?duì)數(shù)化為常用對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件對(duì)數(shù)與對(duì)數(shù)運(yùn)算課件

[一點(diǎn)通]

對(duì)數(shù)式的證明和對(duì)數(shù)式的化簡(jiǎn)的基本思路是一致的，就是根據(jù)對(duì)數(shù)的運(yùn)算