The value of ${\log _2}.{\log _3}....{\log _{100}}{100^{{{99}^{{{98}^{{.^{{.^{{{.2}^1}}}}}}}}}}}$ is

The product of all positive real values of $x$ satisfying the equation $x^{\left(16\left(\log _5 x\right)^3-68 \log _5 x\right)}=5^{-16}$is. . . . .

If ${1 \over 2} \le {\log _{0.1}}x \le 2$ then

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

For $y = {\log _a}x$ to be defined $'a'$ must be

If ${\log _{10}}3 = 0.477$, the number of digits in ${3^{40}}$ is