關于對數運算公式大全,對數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: 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 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1(換掉M和N) a^[log(a)(MN)] = a^[log(a)(M)]×a^[log(a)(N)] 由指數的性質 a^[log(a)(MN)] = a^{[log(a)(M)] + [log(a)(N)]} 又因為指數函數是單調函數,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1(換掉M和N) a^[log(a)(M÷N)] = a^[log(a)(M)]÷a^[log(a)(N)] 由指數的性質 a^[log(a)(M÷N)] = a^{[log(a)(M)] - [log(a)(N)]} 又因為指數函數是單調函數,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1(換掉M) a^[log(a)(M^n)] = {a^[log(a)(M)]}^n 由指數的性質 a^[log(a)(M^n)] = a^{[log(a)(M)]*n} 又因為指數函數是單調函數,所以 log(a)(M^n)=nlog(a)(M)基本性質4推廣log(a^n)(b^m)=m/n*[log(a)(b)]推導如下:由換底公式(換底公式見下面)[lnx是log(e)(x)e稱作自然對數的底] log(a^n)(b^m)=ln(a^n)÷ln(b^n)由基本性質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)] --------------------------------------------(性質及推導 完)其他性質[編輯本段]性質一:換底公式log(a)(N)=log(b)(N)÷log(b)(a)推導如下: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log(b)(b)=1 =1/log(b)(a) 還可變形得: log(a)(b)×log(b)(a)=1。
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