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

Let $\left(x_0, y_0\right)$ be the solution of the following equations $(2 x)^{\ln 2} =(3 y)^{\ln 3}$ $3^{\ln x} =2^{\ln y}$ . Then $x_0$ is

Solution set of inequality ${\log _{10}}({x^2} - 2x - 2) \le 0$ is

The set of real values of $x$ satisfying ${\log _{1/2}}({x^2} - 6x + 12) \ge - 2$ is

If ${\log _{10}}x + {\log _{10}}\,y = 2$ then the smallest possible value of $(x + y)$ is

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