x x and y y

Algebra Level 2

Given x , y ≠ − 1 x,y \neq -1 such that

{ x = y + 1 x + 1 y = x + 1 y + 1 \begin{cases} x = \dfrac{y+1}{x+1} \\ y = \dfrac{x+1}{y+1} \end{cases}

Determine the value of log ⁡ x y \log xy .


The answer is 0.

Wildan Bagus Wicaksono by Wildan Bagus Wicaksono , 18, Indonesia

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2 solutions

Chew-Seong Cheong
Jun 17, 2019

log ⁡ x y = log ⁡ ( y + 1 x + 1 × x + 1 y + 1 ) = log ⁡ 1 = 0 \log xy = \log \left(\dfrac {y+1}{x+1} \times \dfrac {x+1}{y+1} \right) = \log 1 = \boxed 0

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The two equations reduce to the following equation in x :

( x + 1 ) 3 (x+1)^3 (x-1)=0. Since x is not -1, therefore x=1. Hence y=1. So xy=1 and log(xy)=0

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