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

If ${x^{\frac{3}{4}(\log_3 x)^2 + \log_3 x - \frac{5}{4}} = \sqrt{3}}$,then $x$ is:

The number of solutions of the equation $\log _{4}(x-1)=\log _{2}(x-3)$ 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 $\frac{\pi}{4}$.

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

The solution of the equation $\log _{101} \log _{7}(\sqrt{x+7}+\sqrt{x})=0$ is