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

$\sum\limits_{r = 1}^{89} {{{\log }_3}(\tan \,\,{r^o})} = $

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

If ${\log _{1/\sqrt 2 }}\sin x > 0,x \in [0,\,\,4\pi ],$ then the number of values of $x$ which are integral multiples of ${\pi \over 4},$ is

The number of integral solutions $x$ of $\log _{\left(x+\frac{7}{2}\right)}\left(\frac{x-7}{2 x-3}\right)^2 \geq 0$ is

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