己知2㏒mx=㏒kx+㏒nx求证n²=(kn)∧㏒km

Question

己知2㏒m x=㏒k x+㏒n x求证n²=(kn)∧㏒k m

Answer 1

2㏒m x=㏒k x+㏒n x
2lgx/lgm=lgx/lgk+lgx/lgn,【换底公式,loga b=logc b/logc a】
2/lgm=1/lgk+1/lgn
2lgk*lgn=lgm*lgn+lgm*lgk,【两边同×lgk*lgm*lgn】
2lgn=lgm*lgn/lgk+lgm*lgk/lgk
2lgn=(lgm/lgk)*(lgn+lgk)=(lgm/lgk)*lg(nk)=logk m*lg(nk),【loga b+loga c=loga (bc)；换底公式,loga b=logc b/logc a,(lgm/lgk)=logk m】
lgn²/lg(nk)=logk m,【loga b^n=nloga b】
log(nk) n²=logk m,【换底公式,loga b=logc b/logc a；lgn²/lg(nk)=以(nk)为底,n²的对数】
log(kn) n²=logk m,【nk=kn】
∴n²=(kn)^logk m；【根据对数的定义：a^b=N,loga N=b】