换底公式的详细证明？对数函数

令y=log(b)a
则a=b^y
两边取以c为底的对数
log(c)a=log(c)b^y=ylog(c)b
所以y=log(b)a=log(c)a/log(c)b

loa(a,b)=x,
a^x=b,
log(m,a^x)=log(a,b),
xlog(m,a)=log(a,b),
x=loa(a,b)=log(a,b)/log(m,a).

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)

设N＝logab（表示以a为底b的对数）
b＝a^N
lnb=Nlna
N=lnb/lna=logab