Let $S$ be the sum of the digits of the number $15^2 \times 5^{18}$ in base $10$. Then,

The value of $\log_2(\log_3(\dots(\log_{100}(100^{99^{98^{\dots^{2^1}}})))\dots))}$ is

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

Consider all natural numbers whose decimal expansion has only the even digits $0, 2, 4, 6, 8$. Suppose these are arranged in increasing order. If $a_n$ denotes the $n$-th number in this sequence,then $\frac{\lim_{n \rightarrow \infty} \log a_n}{\log n}$ equals

If $x_n > x_{n-1} > \dots > x_2 > x_1 > 1$,then the value of $\log_{x_1} \log_{x_2} \log_{x_3} \dots \log_{x_n} (x_n^{x_{n-1}^{\dots^{x_1}}})$ 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