對數(shù)的性質(zhì)及推導(dǎo)
用^表示乘方,用log(a)(b)表示以a為底,b的對數(shù)
*表示乘號,/表示除號
定義式：
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.這個(gè)就不用推了吧,直接由定義式可得(把定義式中的[n=log(a)(b)]帶入a^n=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?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?yàn)橹笖?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?yàn)橹笖?shù)函數(shù)是單調(diào)函數(shù),所以
log(a)(M^n)=nlog(a)(M)
其他性質(zhì)：
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?yàn)镹=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)
性質(zhì)二：（不知道什么名字）
log(a^n)(b^m)=m/n*[log(a)(b)]
推導(dǎo)如下
由換底公式[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 ）
公式三:
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