Question

plz solve the answer

Answer 1

Let, $\frac{ log(x) }{y - z} = \frac{ log(y) }{z - x} = \frac{ log(z) }{x - y} = k$

Therefore , We can say at present :

log x = k (y-z) = ky - kz

log y = k (z-x) = kz - kx

log z = k (x-y) = kx - ky

Now, We can proceed as under :

$log( {x}^{x} \times {y}^{y} \times {z}^{z} )$

$= log \: {x}^{x} + log \: {y}^{y} + log \: {z}^{z}$

$= x \: log \: x \: + y \: log \: y \: + \: z \: log \: z$

$= \: kxy \: - \: kzx \: + \: kyz \: - \: kyx \: + \: zkx \: - \: kyz$

= 0

= log 1

ULTIMATELY, WE GET :

$log( {x}^{x} \times {y}^{y} \times {z}^{z} ) = \: 0$

$log( {x}^{x} \times {y}^{y} \times {z}^{z} ) = \: log \: 1$

Cancelling "log" from both sides, we get :

$( {x}^{x} \times {y}^{y} \times {z}^{z} ) = 1$ [PROVED]

Answer 2

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