Prove the following:

$(\text{log} \space x)^2 - (\text{log} \space y)^2 = \text{log} \space \dfrac{x}{y}.\text{log} \space xy$

Prove the following:

$2\text{log} \space \dfrac{11}{13} + \text{log} \space \dfrac{130}{77} - \text{log} \space \dfrac{55}{91} = \text{log} \space 2.$

If $\text{log} \space \dfrac{x + y}{2} = \dfrac{1}{2}(\text{log} \space x + \text{log} \space y)$, prove that x = y.

If a, b are positive real numbers, a > b and a² + b² = 27ab, prove that

$\text{log} \space\Big(\dfrac{a - b}{5}\Big) = \dfrac{1}{2}(\text{log} \space a + \text{log} \space b)$