If ${x^y} = {y^x}$,then ${(x/y)^{(x/y)}} = {x^{(x/y) - k}}$,where $k = $

The number of solutions of the equation $\frac{1}{2} \log _{\sqrt{3}}\left(\frac{x+1}{x+5}\right)+\log _{9}(x+5)^{2}=1$ is

If $2 \log (x+1)-\log (x^{2}-1)=\log 2$, then $x=$

If ${a^x} = bc$,${b^y} = ca$,and ${c^z} = ab$,then the value of $xyz$ is equal to:

The number of solutions of the equation $\log _{2}\left(x^{2}+2 x-1\right)=1$ is

Let $a, b, x$ be positive real numbers with $a \neq 1, x \neq 1, ab \neq 1$. Suppose $\log_{a} b = 10$,and $\frac{\log_{a} x \cdot \log_{x}(\frac{b}{a})}{\log_{x} b \cdot \log_{ab} x} = \frac{p}{q}$,where $p$ and $q$ are positive integers which are coprime. Then $p+q$ is