An algebra problem by Renz Mina

Let $u=\sqrt{x+\sqrt{x+\cdots}}$ . Then we have that $u^{u^{u^\cdots}}=2$ . This implies that $2\ln{u}=\ln{2}$ or that is $\ln{u}=\ln{\sqrt{2}}$ . We have that $u=\sqrt{2}$ . So $\sqrt{2}=\sqrt{x+\sqrt{x+\cdots}}$ . This means that $2=x+\sqrt{x+\sqrt{x+\cdots}}=x+\sqrt{2}$ . Thus $x=2-\sqrt{2}=0.59$