If $\log _{1/\sqrt{2}} \sin x > 0$ for $x \in [0, 4\pi]$,then the number of values of $x$ which are integral multiples of $\frac{\pi}{4}$ is:

If $x_n > x_{n-1} > \dots > x_2 > x_1 > 1$,then the value of $\log_{x_1} \log_{x_2} \log_{x_3} \dots \log_{x_n} (x_n^{x_{n-1}^{\dots^{x_1}}})$ is:

$x = \log \left( \frac{1}{y} + \sqrt{1 + \frac{1}{y^2}} \right) \Rightarrow y$ is equal to

Suppose $\log _a b + \log _b a = c$. The smallest possible integer value of $c$ for all $a, b > 1$ is

$A$ real root of the equation $\log_{4}\{\log_{2}(\sqrt{x + 8} - \sqrt{x})\} = 0$ 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