AIME 2018a Problem 5

For each ordered pair of real numbers $(x,y)$ satisfying $$\log_2(2x+y) = \log_4(x^2+xy+7y^2)$$there is a real number $K$ such that $$\log_3(3x+y) = \log_9(3x^2+4xy+Ky^2).$$Find the product of all possible values of $K$.