Let a, b, c be three distinct positive real numbers such that (2 a)^log_e a=(b c)^log_e b and b^log_e 2=a^log_e c. Then 6 a+5 b c is equal to ____.

Let $\displaystyle \mathrm{a}, \mathrm{b}, \mathrm{c}$ be three distinct positive real numbers such that $\displaystyle (2 a)^{\log _e a}=(b c)^{\log _e b}$ and $\displaystyle b^{\log _e 2}=a^{\log _e c}$.Then $\displaystyle 6 a+5 b c$ is equal to $\displaystyle \_\_\_\_$.