最新log計(jì)算知識講解

1、概念如果*x二N(a>0,且a不等于1)，則數(shù)x叫做以a為底N的對數(shù)，記做x二log(a) (N),其中d要寫于log右下?！?對數(shù)性質(zhì)與運(yùn)算法則如下。性質(zhì) loga(l)二 0； loga(d)二1； 負(fù)數(shù)與零無對數(shù).對數(shù)恒等式alogaN=N (a>0 , dHl)運(yùn)算法則 logs(MN)二logaH+logdN； logd (M/N) =logaM-logaN； 對logaM中M的n次方有=nlogaM：如果 Em,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 a"n二b(a>0 且 &H1)則 n=l

2、og(a) (b)基本性質(zhì)：1、*(log(a)(b)=b2、log(a)(MN)二log(a)(M)+log(a)(N);3、log (a) (M4-N)=log(a) (M) -log (a) (N);4、log (a) (M"n)=nlog(a) (M)5、log(a n)M=l/nlog(a) (Nf)推導(dǎo)：1、因?yàn)?n=log(a) (b),代入則 a"n二b,即 (log(a) (b)=bo2、MN二MXN由基本性質(zhì)1(換掉1和2a" log (a) (MN) = a" log (a) (M) Xa" log (a) (N)由指數(shù)的

3、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)二 log (a) (M) + log (a) (N)3、與(2)類似處理M/N二MFN由基本性質(zhì)1(換掉1和2log (a) (M4-N)二 a" log(a) (M) 4-a" log (a) (N)由指數(shù)的性質(zhì)a"log(a) (M4-N) = 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) (MH-N) = log (a) (M) - log (

4、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)l = a"log(a) (M) 由指數(shù)的性質(zhì) a-log (a) (M"n) = log(a) 乂因?yàn)橹笖?shù)函數(shù)是單調(diào)函數(shù)，所以log(a)(M"n)二nlog(a)(M)基本性質(zhì)4推廣log(an)(b m)=m/n*1og(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(an) (b"m)=ln(b"m) 4-ln(an)換底公式的推導(dǎo)：

5、設(shè)X二bF,則log(an) (b"m)=log(ey) (e"x)=x/y x=ln(b'm), y=ln(a"n) 得： log (a"n) (b"m)=ln(b"m) 4-ln(a"n)由基本性質(zhì) 4 可得 log (a n) (b m)二mX In (b) 4- n X In (a)=(m-i-n) X In(b)4-In(a)再由換底公式 log(an) (b"m)=m4-nX log(a) (b)換底公式設(shè) bFm, a二c"n,貝lj b二(c"n) “m二c" (mn)對取以a為底的對數(shù)，有：log(R(b)二m.對取以c為底的對數(shù)，有：log(c)(b)二mn/，得：log(c) (b)/log(a) (b) =n=log(c) (a)log (a) (b)=log(c) (b)/log(c) (a)注：log)(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&為底數(shù)的公式代換：logae=l/ (Ina)!, log c blog a b lo a推導(dǎo)公式log(l/a)(l/b)=loga(b) logs (b)*logb(a)=l求導(dǎo)數(shù)(logax)'=l/x