If x² + y² = 23xy, prove that log$\dfrac{x + y}{5} = \dfrac{1}{2}$(log x + log y).

Given,

x² + y² = 23xy

Above equation can be written as,

$\Rightarrow x^2 + y^2 = 25xy - 2xy \\[1em] \Rightarrow x^2 + y^2 + 2xy = 25xy \\[1em] \Rightarrow \dfrac{x^2 + y^2 + 2xy}{25} = xy \\[1em] \Rightarrow \Big(\dfrac{x + y}{5}\Big)^2 = xy$

Taking log on both sides we get,

$\Rightarrow \text{log }\Big(\dfrac{x + y}{5}\Big)^2 = \text{log }xy \\[1em] \Rightarrow 2\text{log }\dfrac{x + y}{5} = \text{log x + log y} \\[1em] \Rightarrow \text{log }\dfrac{x + y}{5} = \dfrac{1}{2}(\text{log x + log y}) \\[1em]$

Hence, proved that $\text{log}\dfrac{x + y}{5} = \dfrac{1}{2}(\text{log x + log y})$.