Log of a Power

For every $x \ge -\frac{1}{e}$, there is a unique number $W(x) \ge -1$ such that $$W(x) e^{W(x)} = x.$$ The function $W$ is called Lambert's $W$ function. Let $y$ be the unique positive number such that $$\frac{y}{\log_{2} y} = - \frac{3}{5} .$$ The value of $y$ is of the form $e^{-W(z \ln 2)}$ for some rational number $z$. What is the value of $z$? Express your answer as a fraction a/b in simplest terms.

Source: Math Prize for Girls