1、.性質(zhì)loga(1)=0；loga(a)=1；負(fù)數(shù)與零無(wú)對(duì)數(shù).2對(duì)數(shù)恒等式alogaN=N (a0 ，a1）3運(yùn)算法則loga(MN)=logaM+logaN；loga(M/N)=logaMlogaN； 對(duì)logaM中M的n次方有=nlogaM；如果a=em，則m為數(shù)a的自然對(duì)數(shù)，即lna=m，e=2.718281828為自然對(duì)數(shù)的底。定義： 若an=b(a0且a1) 則n=log(a)(b)基本性質(zhì)：1、a(log(a)(b)=b2、log(a)(MN)=log(a)(M)+log(a)(N);3、log(a)(MN)=log(a)(M)-log(a)(N);4、log(a)(Mn)=nlo

2、g(a)(M)5、log(an)M=1/nlog(a)(M)推導(dǎo)：1、因?yàn)閚=log(a)(b)，代入則an=b，即a(log(a)(b)=b。2、MN=MN由基本性質(zhì)1(換掉M和N)alog(a)(MN) = alog(a)(M)alog(a)(N)由指數(shù)的性質(zhì)alog(a)(MN) = alog(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(lèi)似處理 M/N=MN由基本性質(zhì)1(換掉M和N)alog(a)(MN) = alog(a)(M)alog(a)(N)由指數(shù)的性質(zhì)alog(a)(MN) =

3、 alog(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)4、與（2）類(lèi)似處理Mn=Mn 由基本性質(zhì)1(換掉M) alog(a)(Mn) = alog(a)(M)n由指數(shù)的性質(zhì)alog(a)(Mn) = alog(a)(M)*n又因?yàn)橹笖?shù)函數(shù)是單調(diào)函數(shù)，所以log(a)(Mn)=nlog(a)(M)基本性質(zhì)4推廣log(an)(bm)=m/n*log(a)(b)推導(dǎo)如下： 由換底公式（換底公式見(jiàn)下面）lnx是log(e)(x)，e稱(chēng)作自然對(duì)數(shù)的底 log(an)(bm)=ln(bm)ln(an)換底公式的推導(dǎo)

4、： 設(shè)ex=bm,ey=an 則log(an)(bm)=log(ey)(ex)=x/y x=ln(bm),y=ln(an) 得：log(an)(bm)=ln(bm)ln(an)由基本性質(zhì)4可得 log(an)(bm) = mln(b)nln(a) = (mn)ln(b)ln(a)再由換底公式 log(an)(bm)=mnlog(a)(b)4換底公式設(shè)b=am，a=cn，則b=(cn)m=c(mn)對(duì)取以a為底的對(duì)數(shù)，有：log(a)(b)=m.對(duì)取以c為底的對(duì)數(shù)，有：log(c)(b)=mn/，得：log(c)(b)/log(a)(b)=n=log(c)(a)log(a)(b)=log(c)(b)/log(c)(a)注：log(a)(b)表示以a為底x的對(duì)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/（lna） 5推導(dǎo)公式log(1/a)(1/b)=loga(b)loga(b)*logb(a)=16求導(dǎo)數(shù)(xlogax)=l