The value of ${\log _3}\,4{\log _4}\,5{\log _5}\,6{\log _6}\,7{\log _7}\,8{\log _8}\,9$ is

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

Let $\quad \sum \limits_{n=0}^{\infty} \frac{n^3((2 n) !)+(2 n-1)(n !)}{(n !)((2 n) !)}=a e+\frac{b}{e}+c$, where $a, b, c \in Z$ and $e=\sum \limits_{n=0}^{\infty} \frac{1}{n!}$ Then $a^2-b+c$ is equal to $…………….$.

If ${\log _e}\left( {{{a + b} \over 2}} \right) = {1 \over 2}({\log _e}a + {\log _e}b)$, then relation between $a$ and $b$ will be

The value of ${81^{(1/{{\log }_5}3)}} + {27^{{{\log }_{_9}}36}} + {3^{4/{{\log }_{_7}}9}}$ is equal to