If $a, b, c$ are distinct positive numbers, each different from $1$, such that $[{\log _b}a{\log _c}a - {\log _a}a] + [{\log _a}b{\log _c}b - {\log _b}b]$ $ + [{\log _a}c{\log _b}c - {\log _c}c] = 0,$ then $abc =$
$1$
$2$
$3$
None of these
If ${a^2} + 4{b^2} = 12ab,$ then $\log (a + 2b)$ is
If ${x^{{3 \over 4}{{({{\log }_3}x)}^2} + {{\log }_3}x - {5 \over 4}}} = \sqrt 3 $ then $x$ has
The value of $\sqrt {(\log _{0.5}^24)} $ is
If $x = {\log _5}(1000)$ and $y = {\log _7}(2058)$ then
The sum of all the natural numbers for which $log_{(4-x)}(x^2 -14x + 45)$ is defined is -