15. Let $x>1, y>1, S=\min \left\{\log _{x} 2, \log _{2} y\right.$ , $\left.\log _{y}\left(8 x^{2}\right)\right\}$. Then the maximum value of $S$ is $\qquad$ .
15.2.

From the problem, we have $\log _{x} 2 \geqslant S, \log _{2} y \geqslant S, \log _{r}\left(8 x^{2}\right) \geqslant S$, then $S \leqslant \log ,\left(8 x^{2}\right)=\frac{3+2 \log _{2} x}{\log _{2} y}=\frac{3+\frac{2}{\log _{x} 2}}{\log _{2} y} \leqslant \frac{3+\frac{2}{S}}{S}$. Therefore, $S^{3}-3 S-2 \leqslant 0$, which is $(S-2)(S+1)^{2} \leqslant 0$, yielding $S \leqslant 2$.

When $x=\sqrt{2}, y=4$, equality holds.