Logarithms

If \frac{x(y+z-x)}{log(x)}=\frac{y(z+x-y)}{log(y)}=\frac{z(x+y-z)}{log(z)} ,

prove that:

y^z.z^y=z^x.x^z=x^y.y^x

2 Answers

it is quite a long sum...

x(y+z-x)log x = y(z+x-y)log y = z(x+y-z)log z =1k.

thus, log x = k(xy+yz-x²)
log y = k(yz+xy-y²)
log z = k(zx+yz-z²)

R.T.P :- y^z.z^y = z^x.x^z

take log,
L.H.S
z log y + y log z
putting the values of log y and log z,
we get, L.H.S. = 2kxyz

R.H.S will give the same thing..

similarly do the other one.. :)

no use of doin such long sums now dude....

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