For what values of $x$ is the following identity valid and holds? $\tanh^{-1}(x) = \frac{1}{2} \log_e \left( \frac{1+x}{1-x} \right)$.

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 $\log_2 x + \log_x 2 = \frac{10}{3} = \log_2 y + \log_y 2$ and $x \neq y$,then $x + y = $

If $x, y, z \in R^+$ are such that $z > y > x > 1$,$\log_{y}x + \log_{x}y = \frac{5}{2}$ and $\log_{z}y + \log_{y}z = \frac{10}{3}$,then $\log_{x}z$ is equal to

If ${\log _5}a \cdot {\log _a}x = 2$,then the value of $x$ is:

The number of solution$(s)$ of the equation $\ln(\ln x) = \log_x e$ is -