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If x=1+_{p}qr,, y=1+_{q}rp,, z=1+_{r}pq then the value of (xy

Question

If $x = 1 + lo g_{p} q r, y = 1 + lo g_{q} r p, z = 1 + lo g_{r} pq$ then the value of $(x y + yz + z x - x yz)$ is

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x = 1 + lo g_{p} q r, y = 1 + lo g_{q} r p, z = 1 + lo g_{r} pq

then the value of

(x y + yz + z x - x yz)

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Text solutionVerified

x = 1 + lo g_{p} q r = lo g_{p} p + lo g_{p} q r = lo g_{p} pq r

Similarly,

y = lo g_{q} pq r

and

z = lo g_{r} pq r

∴ x y + yz + z x - x yz

= (lo g_{p} pq r) - (lo g_{q} pq r) + (lo g_{q} pq r) . (lo g_{r} pq r) + (lo g_{r} pq r) (lo g_{p} pq r) - (lo g_{p} pq r) (lo g_{q} pq r) (lo g_{r} pq r)

= [\frac{lo g ( pq r )}{lo g p} + \frac{lo g ( pq r )}{lo g q} + ....] - \frac{( lo g ( pq r ) ) ^{3}}{lo g p . lo g q . lo g r}

= [lo g (pq r)]^{2} [\frac{1}{lo g p . lo g q} + \frac{1}{lo g q . lo g r} + \frac{1}{lo g r . lo g p}] - [lo g (pq r)]^{3} [\frac{1}{lo g p . lo g q . lo g r}]

= [lo g (pq r)]^{2} [\frac{lo g p + lo g q + lo g r}{lo g p . lo g q . lo g r}] - [lo g (pq r)]^{3} [\frac{1}{lo g p . lo g q . lo g r}]

= [lo g (pq r)]^{2} [\frac{lo g ( pq r )}{lo g p . lo g q . lo g r}] - [lo g (pq r)]^{3} [\frac{1}{lo g p . lo g q . lo g r}]

= 0

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Question Text	If $x = 1 + lo g_{p} q r, y = 1 + lo g_{q} r p, z = 1 + lo g_{r} pq$ then the value of $(x y + yz + z x - x yz)$ is
Answer Type	Text solution:1
Upvotes	150