The set of real values of $x$ satisfying ${\log _{1/2}}({x^2} - 6x + 12) \ge - 2$ is

For $y = {\log _a}x$ to be defined $'a'$ must be

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

If $a = {\log _{24}}12,\,b = {\log _{36}}24$ and $c = {\log _{48}}36,$ then $1+abc$ is equal to

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

If ${1 \over {{{\log }_3}\pi }} + {1 \over {{{\log }_4}\pi }} > x,$ then $x$ be