设a,b,c都是不等于1的正数,且ab不等于1,求证a^logcb=b^logca

问题描述:

设a,b,c都是不等于1的正数,且ab不等于1,求证a^logcb=b^logca
如题

logb[a^logc(b)]
=logc(b)*logb(a)
=(lgb/lgc)*(lga/lgb)
=lga/lgc
=logc(a)
logb[b^logc(a)]=logc(a)
所以logb[a^logc(b)]=logb[b^logc(a)]
所以a^logc(b)=b^logc(a)