The value of $\sqrt{(\log_{0.5} 4)^2}$ is

If $a < 1$ and $2 \operatorname{Sinh}^{-1}\left(\frac{a}{\sqrt{1-a^2}}\right)=\log \left(\frac{1+x}{1-x}\right)$,then $x=$

Evaluate the sum: $\sum\limits_{k = 1}^n \frac{1}{\log_{2^k}(a)}$

Find the roots of the equation $2^{x + 2} \cdot 27^{x/(x - 1)} = 9$.

Let $n$ be a positive integer such that $\log _2 \log _2 \log _2 \log _2 \log _2(n) < 0 < \log _2 \log _2 \log _2 \log _2(n)$. Let $l$ be the number of digits in the binary expansion of $n$. Then the minimum and the maximum possible values of $l$ are

If $a, b, c$ are distinct positive numbers,each different from $1$,such that $[(\log _b a)(\log _c a) - (\log _a a)] + [(\log _a b)(\log _c b) - (\log _b b)] + [(\log _a c)(\log _b c) - (\log _c c)] = 0$,then $abc =$