對數(shù)計算公式

1、性質(zhì) loga(1)=0 ; loga (a)=1 ; 負數(shù)與零無對數(shù).2對數(shù)恒等式aAlogaN=N (a>0 , a 1)3運算法則 loga(MN)=l ogaM+logaN; loga(M/N)=l ogaM logaN; 對logaM中M的n次方有=nlogaM ;如果a=eAm則m為數(shù)a的自然對數(shù)，即lna=m，e=2.718281828 為自然對數(shù)的底。定義：若 aAn=b(a>0且 a 1)貝S n=log(a)(b)基本性質(zhì)：1、aA(log(a)(b)=b2、log(a)(MN)=l og(a)(M)+l og(a)(N);3、log(a)(M 寧 N)=j(a

2、)(M)-log(a)(N);4、log(a)(MAn)二nl og(a)(M)5、log(aAn)M=1/nl og(a)(M)推導(dǎo)：1、 因為 n=log(a)(b)，代入則 aAn=b，即 aA(log(a)(b)=b。2、MN=M N由基本性質(zhì)1(換掉M和N)aAlog(a)(MN) = aAlog(a)(M)x aAlog(a)(N)由指數(shù)的性質(zhì)aAlog(a)(MN) = a“l(fā)og (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 M/N二葉N由基本性質(zhì)1(換掉M和N)

3、aAlog(a)(M - N) = aAlog (a) (M)- aAlog(a)(N)由指數(shù)的性質(zhì)aAlog(a)(M 寧 N) = a“l(fā)og (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MAn二MAn 由基本性質(zhì) 1(換掉 M) aAlog(a)(Man) aAlog (a) (M/n由指數(shù)的性質(zhì)aAlog(a)(MA n) = aAlog (a) (M)* n又因為指數(shù)函數(shù)是單調(diào)函數(shù)，所以log(a)(MA n)二 nlog(a) (M)基本性質(zhì)4推廣log(aA n)

4、(bAm)二m/n*log (a) (b)推導(dǎo)如下：由換底公式(換底公式見下面)Inx是log(e)(x), e 稱作自然對數(shù)的底log(a"n)(bm)=ln(Zm) In(aAn)換底公式的推導(dǎo)： 設(shè)eAx=bAm,eAy=aAn則Iog(aAn)(bAm)=log(eAy)(eAx)二x/yx=ln(bAm),y=ln(aAn)得：Iog(aA n)(bAm)=l n( bAm) In(aAn)由基本性質(zhì) 4 可得 Iog(aA n)(bAm) = mx In (b) n x In (a)=(m寧 n) x In(b)寧ln(a)再由換底公式 Iog(aAn)(bAm)=m -

5、 nx log(a)(b)4換底公式設(shè)b=aAm,a=cA n,貝 Ub=(cAnFm二cA(mn)對取以a為底的對數(shù)，有：log(a)(b)=m 對取以c為底的對數(shù)，有：log(c)(b)=mn / ， 得：log(c)(b)/log(a)(b)=n=log(c)(a)二log(a) (b)=log(c)(b)/log(c)(a)注：log(a)(b)表示以a為底x的對數(shù)。換底公式拓展：以e為底數(shù)和以a為底數(shù)的公式代換：logae=1/ (Ina )logc675推導(dǎo)公式Iog(1/a)(1/b)=loga(b)loga(b)*logb(a)=16求導(dǎo)數(shù)(xlogax)'=logax+Ina其中，logax中的a為底數(shù)，x為真數(shù);(logax)'=1/x I