2-x-3-y-4-2-y-3-x-6-prove-that-xy-z-z-1-6-z-2-

Question Number 173871 by mathlove last updated on 20/Jul/22

2^{x} \cdot 3^{y} = 4

2^{y} \cdot 3^{x} = 6 p r o v e t h a t x y = z (z + 1)

6^{z} = 2

Commented by dragan91 last updated on 20/Jul/22

\frac{2^{x} 3^{y}}{2^{y} 3^{x}} = \frac{2}{3}

{(\frac{2}{3})}^{x - y} = \frac{2}{3} \Rightarrow x - y = 1 \Rightarrow x = y + 1

2^{y + 1} 3^{y} = 4

2^{y} 3^{y + 1} = 6

6^{y} = 2 \Rightarrow y = \log_{6} 2 \Rightarrow x = \log_{6} 2 + 1

xy = \log_{6} 2 (\log_{6} 2 + 1)

z = \log_{6} 2

xy = z (z + 1) proved