If $\log _{27}(\log _3 x) = \frac{1}{3}$,then the value of $x$ is

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

If $a^2 + 4b^2 = 12ab$,then $\log(a + 2b) =$

$x = \log \left( \frac{1}{y} + \sqrt{1 + \frac{1}{y^2}} \right) \Rightarrow y$ is equal to

Given that $a, b \in \{0, 1, 2, \ldots, 9\}$ with $a+b \neq 0$ and that $\left(a+\frac{b}{10}\right)^x = \left(\frac{a}{10}+\frac{b}{100}\right)^y = 1000$. Then, $\frac{1}{x} - \frac{1}{y}$ is equal to

$7 \log \left( \frac{16}{15} \right) + 5 \log \left( \frac{25}{24} \right) + 3 \log \left( \frac{81}{80} \right)$ is equal to