If $n = 1983!$,then the value of the expression $\frac{1}{\log_2 n} + \frac{1}{\log_3 n} + \frac{1}{\log_4 n} + \dots + \frac{1}{\log_{1983} n}$ is:

The roots of the equation $2^{x + 2} \cdot 27^{x/(x - 1)} = 9$ are given by

Let $a, b, x$ be positive real numbers with $a \neq 1, x \neq 1, ab \neq 1$. Suppose $\log_{a} b = 10$,and $\frac{\log_{a} x \cdot \log_{x}(\frac{b}{a})}{\log_{x} b \cdot \log_{ab} x} = \frac{p}{q}$,where $p$ and $q$ are positive integers which are coprime. Then $p+q$ is

The value of $\left(\frac{1}{\log _{3} 12}+\frac{1}{\log _{4} 12}\right)$ is

The set of real values of $x$ satisfying the inequality $\log_{0.2} \frac{x + 2}{x} \le 1$ is:

If $\log _{e}\left(x^{2}-16\right) \leq \log _{e}(4 x-11)$,then