取對(duì)數(shù)求極限（取對(duì)數(shù)）

關(guān)于取對(duì)數(shù)求極限，取對(duì)數(shù)這個(gè)問(wèn)題很多朋友還不知道，今天小六來(lái)為大家解答以上的問(wèn)題，現(xiàn)在讓我們一起來(lái)看看吧！

1、對(duì)數(shù)的性質(zhì)及推導(dǎo) 用^表示乘方，用log(a)(b)表示以a為底，b的對(duì)數(shù) *表示乘號(hào)。

2、/表示除號(hào) 定義式： 若a^n=b(a>0且a≠1) 則n=log(a)(b) 基本性質(zhì)： 1.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) 1.這個(gè)就不用推了吧，直接由定義式可得(把定義式中的[n=log(a)(b)]帶入a^n=b) 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?yàn)橹笖?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?yàn)橹笖?shù)函數(shù)是單調(diào)函數(shù)。

3、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?yàn)橹笖?shù)函數(shù)是單調(diào)函數(shù)，所以 log(a)(M^n)=nlog(a)(M) 其他性質(zhì)： 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?yàn)镹=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(wèn)看上面的} 所以log(a)(N)=log(b)(N) / log(b)(a) 性質(zhì)二：（不知道什么名字） log(a^n)(b^m)=m/n*[log(a)(b)] 推導(dǎo)如下 由換底公式[lnx是log(e)(x),e稱作自然對(duì)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 ） 公式三: 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(duì)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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