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

$\text{If } \log _{0.2}(x-1) > \log _{0.04}(x+5), \text{ then }$

Let $\log _a b=4$ and $\log _c d=2$,where $a, b, c, d$ are natural numbers. Given that $b-d=7$,the value of $c-a$ 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

If $a^x = b^y = c^z = d^w$, the value of $x\left(\frac{1}{y} + \frac{1}{z} + \frac{1}{w}\right)$ is