怎么证明对数换底公式 如何证明对数换底公式？谢谢
\u8bc1\u660e\u5bf9\u6570\u7684\u6362\u5e95\u516c\u5f0fN
\u8bbey\uff1dloga
y
\u5219a \uff1dN\uff0e
\u4e24\u8fb9\u53d6\u4ee5a\u4e3a\u5e95\u7684\u5bf9\u6570
a N
ylogm \uff1dlogm
N
logm
y\uff1d\uff0d\uff0d\uff0d\uff0d\uff0d
a
logm
N
N logm
\u5373 loga \uff1d\uff0d\uff0d\uff0d\uff0d\uff0d\uff0d
a \uff0e
logm
\u8bbea^b=N\u2026\u2026\u2026\u2026\u2460
\u5219b=logaN\u2026\u2026\u2026\u2026\u2461
\u628a\u2461\u4ee3\u5165\u2460\u5373\u5f97\u5bf9\u6570\u6052\u7b49\u5f0f\uff1a
a^(logaN)=N\u2026\u2026\u2026\u2026\u2462
\u628a\u2462\u4e24\u8fb9\u53d6\u4ee5m\u4e3a\u5e95\u7684\u5bf9\u6570\u5f97
logaN\u00b7logma=logmN
\u6240\u4ee5
logaN=(logmN)/(logma)
\u5e38\u7528\u5bf9\u6570\u3001\u81ea\u7136\u5bf9\u6570\u3001\u4e00\u822c\u5bf9\u6570\u7684\u8bc1\u660e\uff0c\u53c2\u89c1\u4e0b\u56fe\u3002
\u70b9\u51fb\u653e\u5927\uff0c\u518d\u70b9\u51fb\u518d\u653e\u5927\u3002
证明：设 log(a)b=x，
则 a^x=b
两边同时取以n为底的对数，得：
log(n)a^x=log(n)b
xlog(n)a=log(n)b
x=log(n)b/log(n)a
所以 log(a)b=log(n)b/log(n)a。
设t=log(a)b
则有a^t=b
两边取以e为底的对数
tlna=lnb
t=lnb/lna
即是：log(a)b=lnb/lna
换底公式
log<a> b = log<c> b/log<c> a
let
x=log<a> b
a^x =b (1)
let
y= log<c> b/log<c> a
log<c> a^y = log<c> b
a^y = b (2)
from (1) and (2)
a^x = a^y
=> x=y
=>log<a> b = log<c> b/log<c> a
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绛旓細鍥炵瓟锛氫护logaN/logaB=x,鍒檒ogaN=xlogaB=loga(B^x),鈭碞=B^x, 寰楀埌x=logBN,浠ｅ叆logaN/logaB=logBN,
