[tex]S_1=\log_23+\log_34, \; S_2=\sqrt[3]{1+\sqrt[3]{2}+\ldots+\sqrt[3]{1995}+\sqrt[3]{1996}}\\[/tex]

[tex]u=\log_23\in (1,2)\Rightarrow S_1=u+\frac{2}{u}\in[2\sqrt 2,3) \; \forall u\in(1,2)\Rightarrow \left \lfloor S_1 \right \rfloor=2[/tex]

[tex]\int\limits_0^{1997}\sqrt[3]xdx > \sum\limits_{k=1}^{1996}\sqrt[3]k > \int\limits_0^{1996}\sqrt[3]xdx\wedge \frac 341997^{\frac 43} > S_2^3 > \frac 341996^{\frac 43}\\1. \; S_2^3 > \frac 341996^{\frac 43} > \frac 341976^{\frac 43}=12.13^{\frac 43}19^{\frac 43}=26^3\left(\frac{27\times 19^4}{8\times 13^5}\right)^{1/3}\\27\times 19^4=3518667 > 2970344=8\times 13^5\Rightarrow S_2^3 > 26^3\\[/tex][tex]2. \; S_2^3 < \frac 341997^{\frac 43} < \frac 341998^{\frac 43}=\frac 3427^{\frac 43}2^{\frac 43}37^{\frac 43}=27^3\left(\frac{37^4}{3^{12}\times 4}\right)^{1/3}\\37^4=1874161 < 2125764=3^{12}\times 4\Rightarrow S_2^3 < 27^3\\\lfloor S_2\rfloor=26[/tex]