If $A = {\log _2}{\log _2}{\log _4}256 + 2{\log _{\sqrt 2 \,}}\,2,$ then $A$ is equal to

If ${a^2} + 4{b^2} = 12ab,$ then $\log (a + 2b)$ 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

If $3^x=4^{x-1}$, then $x=$

$(A)$ $\frac{2 \log _3 2}{2 \log _3 2-1}$ $(B)$ $\frac{2}{2-\log _2 3}$ $(C)$ $\frac{1}{1-\log _4 3}$ $(D)$ $\frac{2 \log _2 3}{2 \log _2 3-1}$

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

If $x = {\log _b}a,\,\,y = {\log _c}b,\,\,\,z = {\log _a}c$, then $xyz$ is