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

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

If ${a^x} = b,{b^y} = c,{c^z} = a,$ then value of $xyz$ is

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

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

The value of $\sqrt {(\log _{0.5}^24)} $ is