If $x$ satisfies the inequality $\log _{25} x^2 + (\log _5 x)^2 < 2$, then $x$ belongs to

If $\log _{e}\left(x^{2}-16\right) \leq \log _{e}(4 x-11)$, then

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

Let $a, b, c$ be three distinct positive real numbers such that $(2a)^{\ln a} = (bc)^{\ln b}$ and $b^{\ln 2} = a^{\ln c}$. Then $6a + 5bc$ is equal to $........$.

If ${x^y} = {y^x}$,then ${(x/y)^{(x/y)}} = {x^{(x/y) - k}}$,where $k = $

The set of all real values of $x$ for which $f(x) = \log_2(2^x - 2) + \sqrt{1 - x}$ is real is: