${\log _7}{\log _7}\sqrt {7(\sqrt {7\sqrt 7 } )} = $
$3{\log _2}7$
$1 - 3{\log _3}7$
$1 - 3{\log _7}2$
None of these
If ${\log _{12}}27 = a,$ then ${\log _6}16 = $
If ${\log _k}x.\,{\log _5}k = {\log _x}5,k \ne 1,k > 0,$ then $x$ is equal to
The set of real values of $x$ for which ${2^{{{\log }_{\sqrt 2 }}(x - 1)}} > x + 5$ is
The product of all positive real values of $x$ satisfying the equation $x^{\left(16\left(\log _5 x\right)^3-68 \log _5 x\right)}=5^{-16}$is. . . . .
Let $n$ be the smallest positive integer such that $1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n} \geq 4$. Which one of the following statements is true?