a-x-bc-b-y-ca-c-z-ab-Prove-that-x-1-x-y-1-y-z-1-z-2-Without-using-log-a-b-c-

Question Number 191589 by MATHEMATICSAM last updated on 28/Apr/23

a^{x} = b c, b^{y} = c a, c^{z} = a b .

Prove that, \frac{x}{1 + x} + \frac{y}{1 + y} + \frac{z}{1 + z} = 2 .

(Without using \log)

a \neq b \neq c

Commented by mr W last updated on 27/Apr/23

n o t t r u e i f

a = \frac{1}{2}, b = 1, c = 2

x = y = z = - 1

Answered by Rasheed.Sindhi last updated on 26/Apr/23

a^{x} = b c, b^{y} = c a, c^{z} = a b .

Prove that, \frac{x}{1 + x} + \frac{y}{1 + y} + \frac{z}{1 + z} = 2 .

a^{x} b^{y} c^{z} = (b c) (c a) (a b) = a^{2} b^{2} c^{2}

x = 2, y = 2, z = 2

\frac{x}{1 + x} + \frac{y}{1 + y} + \frac{z}{1 + z}

= \frac{2}{1 + 2} + \frac{2}{1 + 2} + \frac{2}{1 + 2}

= \frac{2}{3} + \frac{2}{3} + \frac{2}{3} = \frac{2}{3} \times 3 = 2