If $\log _k x \cdot \log _5 k = \log _x 5$,where $k \ne 1$ and $k > 0$,then $x$ is equal to:

The solution set of the equation $x^{\log_x(1-x)^2} = 9$ is:

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

$A$ real root of the equation $\log_{4}\{\log_{2}(\sqrt{x + 8} - \sqrt{x})\} = 0$ is

If $a, b, c$ are three distinct positive numbers 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 $abc =$

Let $(x_0, y_0)$ be the solution of the following equations: $(2x)^{\ln 2} = (3y)^{\ln 3}$ and $3^{\ln x} = 2^{\ln y}$. Then $x_0$ is