prove that x^(logy-logz).y^(logz-logx).z^(logx-logy)=1

x^ log y- log z + y^ log z - log x + z ^ log x - log y
introducing log to each term,
log x ^ ( log y- log z) + log y^ (log z - log x) + log z ^ (log x - log y)
= log x ( log y- log z) + log y (log z - log x) + log z (log x - log y)
= log x log y - log x log z + log y log z - log x log y + log x log z -log y log z
=0 
= log 1
removing log from both sides we have ,
x^ log y- log z + y^ log z - log x + z ^ log x - log y = 1

 

Regards

Arun (askIITians forum expert)