Logarithm of $32\root 5 \of 4 $ to the base $2\sqrt 2 $ is

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 $................$.

${\log _4}18$ is

Let $a, b, x$ be positive real numbers with $a \neq 1$, $x \neq 1$, ab $\neq 1$. Suppose $\log _{ a } b =10$, and $\frac{\log _{ a } x \log _{ x }\left(\frac{ b }{ a }\right)}{\log _{ x } b \log _{ ab } x }=\frac{ p }{ q }$, where $p$ and $q$ are positive integers which are coprime. Then $p+q$ is

The sum $\sum \limits_{n=1}^{\infty} \frac{2 n^2+3 n+4}{(2 n) !}$ is equal to :

The sum of all the natural numbers for which $log_{(4-x)}(x^2 -14x + 45)$ is defined is -