對(duì)數(shù)計(jì)算公式

1、性質(zhì)①loga(1)=0①loga(1)=0；②loga(a)=1②loga(a)=1；③負(fù)數(shù)與零無(wú)對(duì)數(shù)負(fù)數(shù)與零無(wú)對(duì)數(shù).2對(duì)數(shù)恒等式a^logaN=Na^logaN=N(a0(a0，a≠1a≠1）3運(yùn)算法則①loga(MN)=logaMlogaN；②loga(MN)=logaM－logaN；③對(duì)logaM中M的n次方有次方有=nlogaM；如果a=e^m，則m為數(shù)a的自然對(duì)數(shù)，即lna=m，e=2.718281828…e=2.71828

2、1828…為自然對(duì)數(shù)為自然對(duì)數(shù)的底。定義：若a^n=b(a0且a≠1)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)(M^n)=nlog(a)(M)5、log(a^n)M=1nlog(a)(M)推導(dǎo)：1、因?yàn)閚=log(a)(b)，代入則a^n=b，即a^(log(a)(b))

3、=b。2、MN=MN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(duì)數(shù)的底]log(a^n)(b^m)=ln(b^m)ln(a^n)換底公式的推導(dǎo)：設(shè)e^x=b^me^y=a^n則log(a^n)(b^m)=log(e^y)(e^x)=xyx=ln(b^m)y=ln(a^n)得：log(a^n)(b^m)=ln(b^m)ln(a^n)由基本性質(zhì)4可得log(a^n)(b^m)=[mln(b)][nln(a

4、)]=(mn)[ln(b)][ln(a)]再由換底公式log(a^n)(b^m)=mn[log(a)(b)]4換底公式設(shè)b=a^m，a=c^n，則b=(c^n)^m=c^(mn)………………………………①對(duì)①取以a為底的對(duì)數(shù)，有：log(a)(b)=m……………………………..②對(duì)①取以c為底的對(duì)數(shù)，有：log(c)(b)=mn……………………………③③②，得：log(c)(b)log(a)(b)=n=log(c)(a)∴l(xiāng)og(a)(