$a^{m \log_a n} = ?$

If ${\log _{12}}27 = a,$ then ${\log _6}16 = $

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

If $x = \log_{b}a$,$y = \log_{c}b$,and $z = \log_{a}c$,then the value of $xyz$ is

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 $x$ and $y$ be real numbers such that $x > 2y > 0$ and $2 \log (x - 2y) = \log x + \log y$. Then,the possible value$(s)$ of $\frac{x}{y}$ is/are: