题目

求证：m^log(a)(n)=n^log(a)(m)

提问时间：2020-07-11

答案

由题设知,log(a)m和log(a)n均有意以
观察可知,当等式两边同时取以a为底的对数时是相等的,故可以逆向思维考虑：
[log(a)n]*[log(a)m]=log(a)[m^log(a)(n)]
[log(a)m]*[log(a)n]=log(a)[n^log(a)(m)]
所以log(a)[m^log(a)(n)]=log(a)[n^log(a)(m)]
由对数函数的单调性可知m^log(a)(n)=n^log(a)(m)