2019 AMC 12B Problem 9

Let $t=\log_2 x.$ Then $\log_4 x=\dfrac{t}{2},$ and the sides are $t,\ \dfrac{t}{2},\ 3.$ The triangle inequalities give $t+\dfrac{t}{2}\gt3$ (so $t\gt2$) and $\dfrac{t}{2}+3\gt t$ (so $t\lt6$); the third inequality is automatic. Thus $2\lt\log_2 x\lt6,$ i.e. $4\lt x\lt64.$ The integers $5,6,\ldots,63$ number $59.$ Thus, B is the correct answer.