对数的性质及推导
定义:
若a^n=b(a>0且a≠1)
则n=log(a)(b)
根本性质:
1、a^(log(a)(b))=b
2、log(a)(a^b)=b
3、log(a)(MN)=log(a)(M)+log(a)(N);
4、log(a)(M÷N)=log(a)(M)-log(a)(N);
5、log(a)(M^n)=nlog(a)(M)
6、log(a^n)M=1/nlog(a)(M)
推导
1、由于n=log(a)(b),代入则a^n=b,即a^(log(a)(b))=b.
2、由于a^b=a^b
令t=a^b
所以a^b=t,b=log(a)(t)=log(a)(a^b)
3、MN=M×N
由根本性质1(换掉M和N)
a^[log(a)(MN)] = a^[log(a)(M)]×a^[log(a)(N)]
由指数的性质
a^[log(a)(MN)] = a^{[log(a)(M)] + [log(a)(N)]}
又由于指数函数是单调函数,所以
log(a)(MN) = log(a)(M) + log(a)(N)
4、与(3)相似处置
MN=M÷N
由根本性质1(换掉M和N)
a^[log(a)(M÷N)] = a^[log(a)(M)]÷a^[log(a)(N)]
由指数的性质
a^[log(a)(M÷N)] = a^{[log(a)(M)] - [log(a)(N)]}
又由于指数函数是单调函数,所以
log(a)(M÷N) = log(a)(M) - log(a)(N)
5、与(3)相似处置
M^n=M^n
由根本性质1(换掉M)
a^[log(a)(M^n)] = {a^[log(a)(M)]}^n
由指数的性质
a^[log(a)(M^n)] = a^{[log(a)(M)]*n}
又由于指数函数是单调函数,所以
log(a)(M^n)=nlog(a)(M)
根本性质4推行
log(a^n)(b^m)=m/n*[log(a)(b)]
推导如下:
由换底公式(换底公式见下面)[lnx是log(e)