If $x = \log_{5}(1000)$ and $y = \log_{7}(2058)$,then:

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

The value of $\left(\left(\log _2 9\right)^2\right)^{\frac{1}{\log _2\left(\log _2 9\right)}} \times(\sqrt{7})^{\frac{1}{\log _4 7}}$ is . . . . . . .

The set of real values of $x$ satisfying the inequality $\log_{0.2} \frac{x + 2}{x} \le 1$ is:

If $a, b, c \neq 0$ and belong to the set $\{0, 1, 2, 3, \ldots, 9\}$, then $\log _{10}\left(\frac{a+10 b+10^2 c}{10^{-4} a+10^{-3} b+10^{-2} c}\right)$ is equal to

Let $a = 3 \sqrt{2}$ and $b = \frac{1}{5^{\frac{1}{6}} \sqrt{6}}$. If $x, y \in \mathbb{R}$ are such that $3x + 2y = \log_a(18)^{\frac{5}{4}}$ and $2x - y = \log_b(\sqrt{1080})$,then $4x + 5y$ is equal to: