For the function $y = \log_a x$ to be defined,the base $a$ must be:

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

If the sum of the first $20$ terms of the series $\log _{7^{1/2}} x + \log _{7^{1/3}} x + \log _{7^{1/4}} x + \dots$ is $460$,then $x$ is equal to:

The solution to the equation ${\log _7}{\log _5}(\sqrt {{x^2} + 5 + x} ) = 0$ is:

Let $\phi (x) = x + 2^{\log_x 3} - 3^{\log_x 2}$. Then which of the following is true?

The value of $\frac{\log_{3} 5 \times \log_{25} 27 \times \log_{49} 7}{\log_{81} 3}$ is