\[\begin{aligned} &\ln(x) < 3& \\ \iff &x < \mathrm e^3& \\ &S=\big]0;\mathrm e^3\big[.& \end{aligned}\]

\[\begin{aligned} &2\ln(x)+200 > 0& \\ \iff &\ln(x) > -\frac{200}{2}& \\ \iff &\ln(x) >-100& \\ \iff &x > \mathrm e^{-100}& \\ &S = \big]\mathrm e^{-100};+\infty\big[.& \end{aligned}\]

\[\begin{aligned} &1 - 2\ln(x) \geqslant 0& \\ \iff &-2\ln(x) \geqslant -1& \\ \iff &\ln(x) \leqslant \frac{-1}{-2}& \\ \iff &\ln(x) \leqslant \frac 1 2& \\ \iff &x \leqslant \mathrm e^{1/2}& \\ \iff &x \leqslant \sqrt{\mathrm e}& \\ &S = \big]0;\sqrt{\mathrm e}\big[.& \end{aligned}\]

\[\begin{aligned} &2\ln(x) - 4\ln(x) < 0& \\ \iff &\ln(x) < \frac{4\ln(3)}2& \\ \iff &\ln(x) < 2\ln(3)& \\ \iff &\ln(x) < \ln\left(3^2\right)& \\ \iff &\ln(x) < \ln(9)& \\ \iff &x < 9& \\ &S = \big]0;9\big[.& \end{aligned}\]