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:

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 $y = \log_x x$, then the value of $\left( \frac{dy}{dx} \right)^2$ at $x = e^2$ is

If $\log _{0.04}(x - 1) \ge \log _{0.2}(x - 1)$,then $x$ belongs to the interval:

Consider all natural numbers whose decimal expansion has only the even digits $0, 2, 4, 6, 8$. Suppose these are arranged in increasing order. If $a_n$ denotes the $n$-th number in this sequence,then $\frac{\lim_{n \rightarrow \infty} \log a_n}{\log n}$ equals

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