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

If ${a^x} = bc$,${b^y} = ca$,and ${c^z} = ab$,then find the value of $xyz$.

$\tanh^{-1}(\frac{1}{2}) + \operatorname{coth}^{-1}(3) = $

If $x+\log _{10}\left(1+2^{x}\right)=x \log _{10} 5+\log _{10} 6$, then the value of $x$ is

If $\log _{1/\sqrt{2}} \sin x > 0$ for $x \in [0, 4\pi]$,find the number of values of $x$ that are integer multiples of $\frac{\pi}{4}$.

Let $k>0$ and $t=\operatorname{sech}^{-1}\left(\frac{1}{2}\right)-\operatorname{cosech}^{-1}\left(\frac{3}{k}\right)$. If $3 e^t=2+\sqrt{3}$,then $k=$