The product of all positive real values of $x$ satisfying the equation $x^{(16(\log_5 x)^3 - 68 \log_5 x)} = 5^{-16}$ is:

Let $a, b, x$ be positive real numbers with $a \neq 1, x \neq 1, ab \neq 1$. Suppose $\log_{a} b = 10$,and $\frac{\log_{a} x \cdot \log_{x}(\frac{b}{a})}{\log_{x} b \cdot \log_{ab} x} = \frac{p}{q}$,where $p$ and $q$ are positive integers which are coprime. Then $p+q$ is

If the sum of the first $20$ terms of the series $\log _{7^{1/2}} x + \log _{7^{1/3}} x + \log _{7^{1/4}} x + \dots$ is $460$,then $x$ is equal to:

The value of $\sqrt{(\log_{0.5} 4)^2}$ is

If $a, b, c \neq 0$ and belong to the set $\{0, 1, 2, 3, \ldots, 9\}$, then $\log _{10}\left(\frac{a+10 b+10^2 c}{10^{-4} a+10^{-3} b+10^{-2} c}\right)$ is equal to

$\tanh^{-1}(\frac{1}{2}) + \operatorname{coth}^{-1}(3) = $