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

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

If ${\log _k}x.\,{\log _5}k = {\log _x}5,k \ne 1,k > 0,$ then $x$ is equal to

If $x, y, z \in R^+$ are such that $z > y > x > 1$ , ${\log _y}x + {\log _x}y = \frac{5}{2}$ and ${\log _z}y + {\log _y}z = \frac{{10}}{3}$ then ${\log _x}z$ is equal to

If $log_ab + log_bc + log_ca$ vanishes where $a, b$ and $c$ are positive reals different than unity then the value of $(log_ab)^3 + (log_bc)^3 + (log_ca)^3$ is