Marinada '22

[lang=hr] Slavomir je od malena bio neobično dijete, a to se nije promijenilo ni kada je zavolio matematičke zadatke. Naime, Slavomirovi omiljeni zadaci bili su oni koji uključuju najružnije izraze. Štoviše, Slavomir je toliko volio ružne izraze da je svoje omiljene uvrstio u osobnu kolekciju. Skupio je mnogo zadataka, ali nažalost Slavomir nije ni približno dobar matematičar koliko je spretan kao kolekcionar. Pomozite Slavomiru da upotpuni svoju zbirku ružnih zadataka lijepim (ili ne) rješenjima! \\Za svaki uređeni par realnih brojeva $(x,y)$ koji zadovoljava $$\log_{2}(2x+y)=\log_{4}(x^2+xy+7y^2)$$ postoji realni broj $Z$ takav da vrijedi $$\log_{3}(3x+y)=\log_{9}(3x^2+4xy+Zy^2).$$ \\Koliki je umnožak svih mogućih $Z$? [/lang] [lang=en] From young age, Slavomir has been one unusual child, a fact that hasn't changed when he fell in love with math problems. Slavomir's favourite problems are those that include the ugliest expressions. Moreover, Slavomir likes ugly expressions so much that he decided to make a personal collection. He's collected many problems. Unfortunately Slavomir isn't nearly as good a mathematician as he is a collector. Help him complete his collection of ugly problems with nice (or not) solutions.\\ For every ordered pair of real numbers $(x,y)$ that satisfy $$\log_{2}(2x+y)=\log_{4}(x^2+xy+7y^2)$$ there exists a real number $Z$ such that $$\log_{3}(3x+y)=\log_{9}(3x^2+4xy+Zy^2)$$ hold true. Calculate the product of all possible values of $Z$. [/lang]