If log$\dfrac{x - y}{2} = \dfrac{1}{2}$(log x + log y), prove that x² + y² = 6xy.

Given,

$\Rightarrow \text{log}\dfrac{x - y}{2} = \dfrac{1}{2}(\text{log x + log y}) \\[1em] \Rightarrow 2\text{log}\dfrac{x - y}{2} = \text{log xy} \\[1em] \Rightarrow \Big(\dfrac{x - y}{2}\Big)^2 = xy \\[1em] \Rightarrow (x - y)^2 = 4xy \\[1em] \Rightarrow x^2 + y^2 - 2xy = 4xy \\[1em] \Rightarrow x^2 + y^2 = 6xy.$

Hence, proved that x² + y² = 6xy.