對數(shù)運算公式大全（對數(shù)運算公式大全）

關(guān)于對數(shù)運算公式大全，對數(shù)運算公式大全這個問題很多朋友還不知道，今天小六來為大家解答以上的問題，現(xiàn)在讓我們一起來看看吧！

1、定義： 若a^n=b(a>0且a≠1) 則n=log(a)(b) 基本性質(zhì)： a^(log(a)(b))=b 2、log(a)(MN)=log(a)(M)+log(a)(N); 3、log(a)(M÷N)=log(a)(M)-log(a)(N); 4、log(a)(M^n)=nlog(a)(M) 推導(dǎo) 因為n=log(a)(b)，代入則a^n=b，即a^(log(a)(b))=b。

2、 2、MN=M×N 由基本性質(zhì)1(換掉M和N) a^[log(a)(MN)] = a^[log(a)(M)]×a^[log(a)(N)] 由指數(shù)的性質(zhì) a^[log(a)(MN)] = a^{[log(a)(M)] + [log(a)(N)]} 又因為指數(shù)函數(shù)是單調(diào)函數(shù)，所以 log(a)(MN) = log(a)(M) + log(a)(N) 3、與（2）類似處理 MN=M÷N 由基本性質(zhì)1(換掉M和N) a^[log(a)(M÷N)] = a^[log(a)(M)]÷a^[log(a)(N)] 由指數(shù)的性質(zhì) a^[log(a)(M÷N)] = a^{[log(a)(M)] - [log(a)(N)]} 又因為指數(shù)函數(shù)是單調(diào)函數(shù)，所以 log(a)(M÷N) = log(a)(M) - log(a)(N) 4、與（2）類似處理 M^n=M^n 由基本性質(zhì)1(換掉M) a^[log(a)(M^n)] = {a^[log(a)(M)]}^n 由指數(shù)的性質(zhì) a^[log(a)(M^n)] = a^{[log(a)(M)]*n} 又因為指數(shù)函數(shù)是單調(diào)函數(shù)，所以 log(a)(M^n)=nlog(a)(M)基本性質(zhì)4推廣log(a^n)(b^m)=m/n*[log(a)(b)]推導(dǎo)如下：由換底公式（換底公式見下面）[lnx是log(e)(x)e稱作自然對數(shù)的底] log(a^n)(b^m)=ln(a^n)÷ln(b^n)由基本性質(zhì)4可得log(a^n)(b^m) = [n×ln(a)]÷[m×ln(b)] = (m÷n)×{[ln(a)]÷[ln(b)]}再由換底公式log(a^n)(b^m)=m÷n×[log(a)(b)] --------------------------------------------（性質(zhì)及推導(dǎo) 完）其他性質(zhì)[編輯本段]性質(zhì)一：換底公式log(a)(N)=log(b)(N)÷log(b)(a)推導(dǎo)如下：N = a^[log(a)(N)]a = b^[log(b)(a)]綜合兩式可得N = {b^[log(b)(a)]}^[log(a)(N)] = b^{[log(a)(N)]*[log(b)(a)]}又因為N=b^[log(b)(N)]所以 b^[log(b)(N)] = b^{[log(a)(N)]*[log(b)(a)]}所以 log(b)(N) = [log(a)(N)]*[log(b)(a)] {這步不明白或有疑問看上面的}所以log(a)(N)=log(b)(N) / log(b)(a)公式二：log(a)(b)=1/log(b)(a)證明如下：由換底公式 log(a)(b)=log(b)(b)/log(b)(a) ----取以b為底的對數(shù)log(b)(b)=1 =1/log(b)(a) 還可變形得: log(a)(b)×log(b)(a)=1。

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