求对数函数公式的推导

Question

如题所述

Answer 1

设log(a)(M^n)=y,
则a^y=M^n
M=a^(y/n),代入
nlog(a)(M)
=nlog(a)a^(y/n)
=n·y/n
=y.
∴log(a)(M^n)=nlog(a)(M)
.
log(a)(N)=log(b)(N)
/
log(b)(a)是换底公式.
令t=log(a)(N),
则a^t=N,
两边取以b为底的对数,
log(b)a^t=log(b)N,
t=log(b)(N)
/
log(b)(a).
∴log(a)(N)=log(b)(N)
/
log(b)(a).
说明:对数式是用指数式来定义的,故常常将它们互化.
可以看出这两个证明都是转化成指数式来证明的.

Answer 2

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)]
所以log(a)(n)=log(b)(n)
/
log(b)(a)