The set of real values of $x$ for which ${\log _{0.2}}{{x + 2} \over x} \le 1$ is

If $n = 1983!$, then the value of expression $\frac{1}{{{{\log }_2}n}} + \frac{1}{{{{\log }_3}n}} + \frac{1}{{{{\log }_4}n}} + ....... + \frac{1}{{{{\log }_{1983}}n}}$ is equal to

Let $a , b , c$ be three distinct positive real numbers such that $(2 a)^{\log _{\varepsilon} a}=(b c)^{\log _e b}$ and $b^{\log _e 2}=a^{\log _e c}$. Then $6 a+5 b c$ is equal to $........$.

If ${\log _{10}}x = y,$ then ${\log _{1000}}{x^2} $ is equal to

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

The number of integral solutions $x$ of $\log _{\left(x+\frac{7}{2}\right)}\left(\frac{x-7}{2 x-3}\right)^2 \geq 0$ is