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

The equation $\log_e x + \log_e(1 + x) = 0$ can be written as

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

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

If $\log _3 x+\log _3 y=2+\log _3 2$ and $\log _3(x+y)=2$, then

If ${x_n} > {x_{n - 1}} > ... > {x_2} > {x_1} > 1$,then the value of ${\log _{{x_1}}}{\log _{{x_2}}}{\log _{{x_3}}}.....{\log _{{x_n}}}({x_n}^{{x_{n - 1}}^{{.^{{.^{{.^{{x_1}}}}}}}}})$ is equal to