If $\log _{1/\sqrt{2}} \sin x > 0$ for $x \in [0, 4\pi]$,then the number of values of $x$ which are integral multiples of $\frac{\pi}{4}$ is:

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

The value of $x$ satisfying $\log _a x + \log _{\sqrt{a}} x + \log _{\sqrt[3]{a}} x + \dots + \log _{\sqrt[n]{a}} x = \frac{n(n+1)}{2}$ is given by the sum of $n$ terms. If the series is $\log _a x + \log _{a^{1/2}} x + \log _{a^{1/3}} x + \dots + \log _{a^{1/n}} x = S$,find $x$ for the given expression $\log _a x + \log _{\sqrt{a}} x + \dots + \log _{a^{1/n}} x = \frac{n(n+1)}{2}$. For the specific case provided: $\log _a x + \log _{a^{1/2}} x + \dots + \log _{a^{1/n}} x = \sum_{k=1}^{n} k \log_a x = \frac{n(n+1)}{2} \log_a x$. Given $\frac{n(n+1)}{2} \log_a x = \frac{a+1}{2}$,find $x$.

If $x$ satisfies the inequality $\log _{25} x^2 + (\log _5 x)^2 < 2$, then $x$ belongs to

If $x, y, z \in R^+$ are such that $z > y > x > 1$,$\log_{y}x + \log_{x}y = \frac{5}{2}$ and $\log_{z}y + \log_{y}z = \frac{10}{3}$,then $\log_{x}z$ is equal to

The set of all real values of $x$ for which $f(x) = \log_2(2^x - 2) + \sqrt{1 - x}$ is real is: