跪求对数函数运算公式?
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跪求对数函数运算公式?
![跪求对数函数运算公式?](/uploads/image/z/5174390-38-0.jpg?t=%E8%B7%AA%E6%B1%82%E5%AF%B9%E6%95%B0%E5%87%BD%E6%95%B0%E8%BF%90%E7%AE%97%E5%85%AC%E5%BC%8F%3F)
基本性质:
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、因为n=log(a)(b),代入则a^n=b,即a^(log(a)(b))=b.
2、因为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
由基本性质1(换掉M和N)
a^[log(a)(MN)] = a^[log(a)(M)]×a^[log(a)(N)] =(M)*(N)
由指数的性质
a^[log(a)(MN)] = a^{[log(a)(M)] + [log(a)(N)]}
两种方法只是性质不同,采用方法依实际情况而定
又因为指数函数是单调函数,所以
log(a)(MN) = log(a)(M) + log(a)(N)
4、与(3)类似处理
MN=M÷N
由基本性质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)]}
又因为指数函数是单调函数,所以
log(a)(M÷N) = log(a)(M) - log(a)(N)
5、与(3)类似处理
M^n=M^n
由基本性质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}
又因为指数函数是单调函数,所以
log(a)(M^n)=nlog(a)(M)
基本性质4推广
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)(x),e称作自然对数的底]
log(a^n)(b^m)=ln(b^m)÷ln(a^n)
换底公式的推导:
设e^x=b^m,e^y=a^n
则log(a^n)(b^m)=log(e^y)(e^x)=x/y
x=ln(b^m),y=ln(a^n)
得:log(a^n)(b^m)=ln(b^m)÷ln(a^n)
由基本性质4可得
log(a^n)(b^m) = [m×ln(b)]÷[n×ln(a)] = (m÷n)×{[ln(b)]÷[ln(a)]}
再由换底公式
log(a^n)(b^m)=m÷n×[log(a)(b)]
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、因为n=log(a)(b),代入则a^n=b,即a^(log(a)(b))=b.
2、因为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
由基本性质1(换掉M和N)
a^[log(a)(MN)] = a^[log(a)(M)]×a^[log(a)(N)] =(M)*(N)
由指数的性质
a^[log(a)(MN)] = a^{[log(a)(M)] + [log(a)(N)]}
两种方法只是性质不同,采用方法依实际情况而定
又因为指数函数是单调函数,所以
log(a)(MN) = log(a)(M) + log(a)(N)
4、与(3)类似处理
MN=M÷N
由基本性质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)]}
又因为指数函数是单调函数,所以
log(a)(M÷N) = log(a)(M) - log(a)(N)
5、与(3)类似处理
M^n=M^n
由基本性质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}
又因为指数函数是单调函数,所以
log(a)(M^n)=nlog(a)(M)
基本性质4推广
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)(x),e称作自然对数的底]
log(a^n)(b^m)=ln(b^m)÷ln(a^n)
换底公式的推导:
设e^x=b^m,e^y=a^n
则log(a^n)(b^m)=log(e^y)(e^x)=x/y
x=ln(b^m),y=ln(a^n)
得:log(a^n)(b^m)=ln(b^m)÷ln(a^n)
由基本性质4可得
log(a^n)(b^m) = [m×ln(b)]÷[n×ln(a)] = (m÷n)×{[ln(b)]÷[ln(a)]}
再由换底公式
log(a^n)(b^m)=m÷n×[log(a)(b)]