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

The set of real values of $x$ for which ${2^{{{\log }_{\sqrt 2 }}(x - 1)}} > x + 5$ is

$\sum\limits_{n = 1}^n {{1 \over {{{\log }_{{2^n}}}(a)}}} = $

If ${\log _7}2 = m,$ then ${\log _{49}}28$ is equal to

If ${\log _{10}}3 = 0.477$, the number of digits in ${3^{40}}$ 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 $........$.