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:

If $x+\log _{10}\left(1+2^{x}\right)=x \log _{10} 5+\log _{10} 6$, then the value of $x$ is

If $a = \log_{24} 12, b = \log_{36} 24$ and $c = \log_{48} 36,$ then $1 + abc$ is equal to

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

If $\log_{a}b + \log_{b}c + \log_{c}a = 0$,where $a, b,$ and $c$ are positive real numbers different from $1$,then the value of $(\log_{a}b)^3 + (\log_{b}c)^3 + (\log_{c}a)^3$ 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