log x/y-z = log y/z-x = log z/x-y prove that x^x y^y z^z = 1
Answers
Answer:
HELLO DEAR,
it seems there is some mistakes in QUESTIONS,
the correct question is If logx/y-z = logy/z-x = logz/x-y ,show that :- \bold{x^x * y^y *z^z = 1}x
x
∗y
y
∗z
z
=1
let \bold{log\frac{ x}{y - z} =log \frac{ y}{ z - x} = log\frac{z}{x - y} = K}log
y−z
x
=log
z−x
y
=log
x−y
z
=K
so,
logx = (y - z)k ,
logy = (z - x)k ,
logz = (x - y)k
now,
We have to prove,
\bold{x^x*y^y*z^z = 1}x
x
∗y
y
∗z
z
=1
let \bold{x^x*y^y*z^z = P}x
x
∗y
y
∗z
z
=P
\bold{log(x^x *y^y*z^z) = log P}log(x
x
∗y
y
∗z
z
)=logP
\bold{xlogx + ylogy + zlogz = log P}xlogx+ylogy+zlogz=logP
now, using the above values,
\bold{x(y - z)k + y(z - x)k + z(x - y)k = log P}x(y−z)k+y(z−x)k+z(x−y)k=logP
\bold{xky - xkz + ykz - xky + xkz - zky = log P}xky−xkz+ykz−xky+xkz−zky=logP
\bold{0 = log P}0=logP
\bold{log1 = logP}log1=logP
HENCE, p = 1
Thus , \bold{x^x*y^y*z^z = 1}x
x
∗y
y
∗z
z
=1
I HOPE ITS HELP YOU DEAR,
THANKS