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

If $a, b, c$ are three distinct positive numbers different from $1$ such that $[(\log_b a)(\log_c a) - \log_a a] + [(\log_a b)(\log_c b) - \log_b b] + [(\log_a c)(\log_b c) - \log_c c] = 0$,then $abc =$

If $\alpha = \log_e(2+\sqrt{3})$,then $\frac{\cosh \alpha}{1-\tanh \alpha} + \frac{\sinh \alpha}{1-\coth \alpha} = $

Find the solution of the equation $\log_{7}(\log_{5}(\sqrt{x^2 + x + 5})) = 0$.

The value of $\{x \in R \mid \log_{10} ((1.6)^{1-x^2} - (0.625)^{6(1+x)}) \in R\}$ is