Question

If log x/(y - z)  =  log y/(z - x)  =  log z/(x - y), then prove that xyz  =  1​

Answer 1

It seems there is some mistakes in question,

the correct question is If log x/y-z = log y/z-x = log z/x-y ,show that :-

x^x × y^y × z^z =1

let,

log x/y-z=log y/z-x=log z/x-y=K

so,

logx = (y - z)k ,

logy = (z - x)k ,

logz = (x - y)k

now,

We have to prove,

x^x × y^y × z^z =1

let,

x^x × y^y × z^z =P

log(x^x × y^y × z^z)=logP

xlogx + ylogy + zlogz =logP

now, using the above values,

x(y-z)K+y(z-x)K+z(x-y)K=logP

xKy-xKz+yKz-xKy+xKz-zKy = logP

0=logP

log1=logP

HENCE, p = 1

Thus,

x^x × y^y × z^z =1

Hope it helps you,

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