Explicație pas cu pas:
a) 14 < 2 la puterea n <129
[tex]14 < {2}^{n} < 129[/tex]
[tex]8 = {2}^{3} < 14 < {2}^{4} = 16 = > 3 < n[/tex]
[tex]128 = {2}^{7} < 129 < {2}^{8} = 256 = > n < 8 \\ [/tex]
[tex]3 < n < 8 = > n \in \{4; 5; 6; 7\}[/tex]
b) 3 la puterea n + 1 <57 < 3 la puterea n + 1
[tex]{3}^{n + 1} < 57 < {3}^{n + 1} [/tex]
n ∈ Ø
c) 3 la puterea n + 1 <57 < 3 la puterea n + 2
[tex]{3}^{n + 1} < 57 < {3}^{n + 2}[/tex]
[tex]27 = {3}^{3} < 57 < {3}^{4} = 81 \\ n + 1 < 4 = > n < 3 \\ n + 2 > 3 = > n > 1 \\ 1 < n < 3 = > n = 2[/tex]
d) 7 la puterea n - 2 <_ 343
[tex] {7}^{n - 2} \leqslant 343[/tex]
[tex]343 = {7}^{3} \\ n - 2 \leqslant 3 = > n \leqslant 5 \\ = > n \in \{0; 1; 2; 3; 4; 5\}[/tex]
