If $A = \log_2 \log_2 \log_4 256 + 2 \log_{\sqrt{2}} 2$,then the value of $A$ is:

The sum of the roots of the equation $x+1-2 \log _{2}\left(3+2^{x}\right)+2 \log _{4}\left(10-2^{-x}\right)=0$ is:

If $\log _{27}(\log _3 x) = \frac{1}{3}$,then the value of $x$ is

The solution of the equation $\log _{101} \log _{7}(\sqrt{x+7}+\sqrt{x})=0$ is

Given that $a, b \in \{0, 1, 2, \ldots, 9\}$ with $a+b \neq 0$ and that $\left(a+\frac{b}{10}\right)^x = \left(\frac{a}{10}+\frac{b}{100}\right)^y = 1000$. Then, $\frac{1}{x} - \frac{1}{y}$ is equal to

If $\log_{a}b + \log_{b}c + \log_{c}a = 0$,where $a, b,$ and $c$ are positive real numbers different from $1$,then the value of $(\log_{a}b)^3 + (\log_{b}c)^3 + (\log_{c}a)^3$ is