If ${\log _4}5 = a$ and ${\log _5}6 = b,$ then ${\log _3}2$ is equal to

Let $S$ be the sum of the digits of the number $15^2 \times 5^{18}$ in base $10$. Then,

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

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

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

The number of solution pairs $(x, y)$ of the simultaneous equations $\log _{1 / 3}(x+y)+\log _3(x-y)=2$ $2^{y^2}=512^{x+1}$ is