If ${a^2} + 4{b^2} = 12ab,$ then $\log (a + 2b)$ is
${1 \over 2}[\log a + \log b - \log 2]$
$\log {a \over 2} + \log {b \over 2} + \log 2$
${1 \over 2}[\log a + \log b + 4\log 2]$
${1 \over 2}[\log a - \log b + 4\log 2]$
The set of real values of $x$ for which ${2^{{{\log }_{\sqrt 2 }}(x - 1)}} > x + 5$ is
The value of ${\log _3}\,4{\log _4}\,5{\log _5}\,6{\log _6}\,7{\log _7}\,8{\log _8}\,9$ 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. . . . .
$\sum\limits_{n = 1}^n {{1 \over {{{\log }_{{2^n}}}(a)}}} = $
Let $\log _a b=4, \log _c d=2$, where $a, b, c, d$ are natural numbers. Given that $b-d=7$, the value of $c-a$ is