lg10=1lg1=0其他1、a^(log(a)(b))=b2、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1、因為n=log(a)(b)，代入則a^n=b，即a^(log(a)(b))=b。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)]}又因為指數函式是單調函式，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)]}又因為指數函式是單調函式，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}又因為指數函式是單調函式，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)