求对数函数公式的推导

log(a)(M^n)
=log(a)(M*M*M*..M) (n个M)
=log(a)（M）+log(a)（M）+log(a)（M）+。。。+log(a)（M）（n个log(a)（M））
=nlog(a)(M)
log(a)(N)
=log(a)[b^log(b)(N)]
=log(b)(N) log(a)(b)
=log(b)(N) log(a)(a^log(b)(a))
=log(b)(N) / log(b)(a)

log(a)(M^n)=nlog(a)(M) 等价于a^(nlogaM)=M^n显然成立
log(a)(N)=log(b)(N) / log(b)(a) 等价于logb(N)=log(a)(N)log(b)(a)
等价于b^[log(a)(N)*log(b)(a)]=N
即a^log(a)(N)=N
显然成立，得证。

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.