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

The equation $x^{\frac{3}{4}(\log_2 x)^2 + \log_2 x - \frac{5}{4}} = \sqrt{2}$ has

The value of $\left(\frac{1}{\log _{3} 12}+\frac{1}{\log _{4} 12}\right)$ 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 $\frac{\log x}{b - c} = \frac{\log y}{c - a} = \frac{\log z}{a - b}$,then which of the following is true?

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