换底公式的详细证明？对数函数

显示全部楼层 · 2019-12-31 14:22:15

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

千问 · 2019-12-31 14:22:15

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

千问 · 2019-12-31 14:22:15

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)

千问 · 2019-12-31 14:22:15

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).