Question

对数恒等式的问题

Answer 1

用^表示乘方，用log(a)(b)表示以a为底，b的对数
*表示乘号，/表示除号
定义式：
若a^n=b(a>0且a≠1)
则n=log(a)(b)
基本性质：
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)
推导
1.这个就不用推了吧，直接由定义式可得(把定义式中的[n=log(a)(b)]带入a^n=b)
2.
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)]
由指数的性质
a^[log(a)(mn)]
=
a^{[log(a)(m)]
+
[log(a)(n)]}
又因为指数函数是单调函数，所以
log(a)(mn)
=
log(a)(m)
+
log(a)(n)
3.与2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)
4.与2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)
其他性质：
性质一：换底公式
log(a)(n)=log(b)(n)
/
log(b)(a)
推导如下
n
=
a^[log(a)(n)]
a
=
b^[log(b)(a)]
综合两式可得
n
=
{b^[log(b)(a)]}^[log(a)(n)]
=
b^{[log(a)(n)]*[log(b)(a)]}
又因为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)]
{这步不明白或有疑问看上面的}
所以log(a)(n)=log(b)(n)
/
log(b)(a)
性质二：（不知道什么名字）
log(a^n)(b^m)=m/n*[log(a)(b)]
推导如下
由换底公式[lnx是log(e)(x),e称作自然对数的底]
log(a^n)(b^m)=ln(a^n)
/
ln(b^n)
由基本性质4可得
log(a^n)(b^m)
=
[n*ln(a)]
/
[m*ln(b)]
=
(m/n)*{[ln(a)]
/
[ln(b)]}
再由换底公式
log(a^n)(b^m)=m/n*[log(a)(b)]