定義：
若a^n=b(a>0且a≠1)
則n=log(a)(b)
基本性質(zhì)：
1、a^(log(a)(b))=b
2、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(dǎo)
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(zhì)1(換掉M和N)
a^[log(a)(MN)] = a^[log(a)(M)]×a^[log(a)(N)]
由指數(shù)的性質(zhì)
a^[log(a)(MN)] = a^{[log(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
MN=M÷N
由基本性質(zhì)1(換掉M和N)
a^[log(a)(M÷N)] = a^[log(a)(M)]÷a^[log(a)(N)]
由指數(shù)的性質(zhì)
a^[log(a)(M÷N)] = a^{[log(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
M^n=M^n
由基本性質(zhì)1(換掉M)
a^[log(a)(M^n)] = {a^[log(a)(M)]}^n
由指數(shù)的性質(zhì)
a^[log(a)(M^n)] = a^{[log(a)(M)]*n}
又因為指數(shù)函數(shù)是單調(diào)函數(shù),所以
log(a)(M^n)=nlog(a)(M)
基本性質(zhì)4推廣
log(a^n)(b^m)=m/n*[log(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(a^n)(b^m)=ln(a^n)÷ln(b^n)
由基本性質(zhì)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(zhì)及推導(dǎo) 完）
其他性質(zhì)
[編輯本段]
性質(zhì)一：換底公式
log(a)(N)=log(b)(N)÷log(b)(a)
推導(dǎo)如下：
N = a^[log(a)(N)]
a = b^[log(b)(a)]
綜合兩式可得
N = {b^[log(b)(a)]}^[log(a)(N)] = b^{[log(a)(N)]*[log(b)(a)]}
又因為N=b^[log(b)(N)]
所以 b^[log(b)(N)] = b^{[log(a)(N)]*[log(b)(a)]}
所以 log(b)(N) = [log(a)(N)]*[log(b)(a)] {這步不明白或有疑問看上面的}
所以log(a)(N)=log(b)(N) / log(b)(a)
公式二：log(a)(b)=1/log(b)(a)
證明如下：
由換底公式 log(a)(b)=log(b)(b)/log(b)(a) ----取以b為底的對數(shù)
log(b)(b)=1 =1/log(b)(a) 還可變形得:log(a)(b)×log(b)(a)=1