Zad 5.10, 5.11, 5.12 oprócz zaznaczonych podpunktów  

zadanie 5.10 a) [latex]7^{x + 2} + 2*7^{x - 1} = 345\ 7^{x-1}(7^3 + 2) = 345\ 7^{x-1} = frac{345}{343 + 2} = 1 = 7^0\ x = 1 [/latex]   b) [latex]3^{2x - 1} + 3^{2x - 2} - 3^{2x - 4} = 315\ 3^{2x - 4}(3^3 + 3^2 - 3^0) = 3^{2x - 4}(27 + 9 - 1) = 3^{2x - 4}*35 = 315\ 3^{2x - 4} = frac{315}{35} = 9 = 3^2\ x = 3[/latex]   c) [latex]2^{x+1} + 3*2^{x-1}-5*2^x+6 = 0\ 2^{x-1}(2^2 + 3 - 5*2) = 2^{x-1}(4 +3-10) = 2^{x-1}*(-3) = - 6\ 2^{x-1} = frac{-6}{-3} = 2 = 2^1\ x = 2[/latex]   d) [latex]4^{2x + 1} = 65*4^{x - 1} - 1\ 4*4^{2x} - frac{65}{4}4^x + 1 = 0\ 16*4^{2x} - 65*4^x + 4 = 0\ Delta = 4225 - 256 = 3969 = 63^2\ 4^{x_1} = frac{65 - 63}{32} = frac{1}{16} = 4^{-2} Rightarrow x_1 = - 2\ 4^{x_2} = frac{65 + 63}{32} = frac{128}{32} = 4^{1} Rightarrow x_1 = 1[/latex]   e) [latex]6^{1 + x} + 6^{1 - x} = 37 |*6^x \ 6*6^{2x} - 37*6^x + 6 = 0\ Delta = 1369 - 144 = 1225 = 35^2\ 6^{x_1} = frac{37 - 35}{12} = frac{1}{6} = 6^{-1} Rightarrow x_1 = -1\ 6^{x_2} = frac{37 + 35}{12} = 6 = 6^{1} Rightarrow x_2 = 1[/latex]   f) [latex]4^{x+1} + 16^{x - 1} = 1536\ frac{1}{16}4^{2x} + 4*4^x - 1536 = 0\ 4^{2x} + 64*4^x - 16*1536 = 0\ Delta = 64^2 + 64*1536 = 64^2(1 + 24) = (64*5)^2 = 320^2\ 4^x > 0\ 4^{x} = frac{-64 + 320}{2} = 128 = 2^7 = 4^{frac{7}{2}}\ x = frac{7}{2} = 3 frac{1}{2}[/latex]   g) [latex]3^{x+2} + 9^{x+1} = 810\ 9*3^{2x} + 9*3^x - 810 = 0\ 3^{2x} + 3^x - 90 = 0\ (3^x - 9)(3^x + 10) = 0\ 3^x = 9 = 3^2\ x = 2[/latex]   h) [latex]frac{8}{3}3^{x-1} + 1 = 9^{x-1}\ frac{1}{9}3^{2x} - frac{8}{9}3^x - 1 = 0\ 3^{2x} - 8*3^x - 9 = 0\ (3^x - 9)(3^x + 1) = 0\ 3^x = 9 = 3^2\ x = 2[/latex]   j) [latex]7^{2x} + 7^x - 686 = 36*7^x\ 7^{2x} - 35*7^x - 686 = 0\ Delta = 1225 + 2744 = 3969 = 63^2\ 7^x > 0\ 7^x = frac{35 + 63}{4} = frac{49}{2}\ x = log_7frac{49}{2} = log_749-log_72 = 2 - log_72[/latex]     zadanie 5.11 a) [latex]5^{x - 3} = 7^{3 - x} |*7^{x - 3}\ (5*7)^{x - 3} = 7^0 = 1 = (5*7)^0\ x = 3[/latex]   b) [latex]13^{x - 5} = 11^{x - 5} |:11^{x - 5}\ left(frac{13}{11} ight)^{x - 5} = 11^0 = 1 = left(frac{13}{11} ight)^0\ x = 5[/latex]   c) [latex]21^{2x+4} = 7^{3x}*3^{4x - 4} |: 7^{3x}*3^{4x - 4}\ frac{21^{2x + 4}}{7^{3x}*3^{4x - 4}} = frac{7^{2x + 4}3^{2x + 4}}{7^{3x}*3^{4x - 4}} = 7^{4 - x}*3^{8 - 2x} = (7*9)^{4 - x} = 7^0 = 1 = (7*9)^0\ x = 4[/latex]   d) [latex]2^{3x}*7^{x - 2} = 4^{x+1} = 2^{2x+2} |:2^{2x+2}\ 2^{x-2}*7^{x-2} = 14^{x-2} = 4^0 = 1 = 14^0\ x = 2[/latex]   e) [latex]12^{2x+4} = 4^{3x}*3^{x+8} |:4^{3x}*3^{x+8}\ frac{12^{2x+4}}{4^{3x}*3^{x+8}} = frac{4^{2x+4}3^{2x+4}}{4^{3x}*3^{x+8}} = 4^{4 - x}*3^{x - 4} = left(frac{4}{3} ight)^{4 - x} = 1\x = 4[/latex]     zadanie 5.12 a) [latex]D: x ot in {3, 7}\ 32^{frac{x+5}{x - 7}} = 0,25 * 128^{frac{x + 17}{x-3}}\ 2^{frac{5(x+5)}{x - 7}} = 2^{frac{7(x + 17)}{x-3} - 4}\ frac{5(x+5)}{x - 7} = frac{7(x + 17)}{x-3} - 4 |*(x - 7)(x - 3)\ 5(x+5)(x - 3) = 7(x+17)(x - 7) - 4(x - 7)(x - 3)\ 5x^2 + 10x - 75 = 7x^2 + 70x - 833 - 4x^2 + 40x - 84\ 2x^2 - 100x + 842 = 0\ x^2 - 50x + 421 = 0\ Delta = 2500 - 1684 = 16*51\ x_1 = frac{50 + 4sqrt{51}}{2} = 25 + 2sqrt{51}\ x_2 = frac{50 - 4sqrt{51}}{2} = 25 - 2sqrt{51}[/latex]   b) [latex]frac{12^{x^2 + 4}}{144^{4x}} = frac{1}{1728}\ frac{12^{x^2 + 4}}{12^{8x}} = frac{1}{1728}\ 12^{x^2 - 8x + 4} = 12^{-3}\ x^2 - 8x + 7 = 0\ (x - 7)(x - 1) = 0\ x = 7 vee x = 1[/latex]   c) [latex]D: x eq - 2\ 9^{x+2} = 3*729^{frac{1}{x + 2}}\ 3^{2x+4} = 3^{frac{6}{x + 2} + 1}\ 2x+4 = frac{6}{x + 2} + 1\ 2x + 3 = frac{6}{x + 2}\ (2x + 3)(x + 2) = 6\ 2x^2 + 7x = 0\ x(x + 3,5) =0\ x = 0 vee x = - 3,5[/latex]   d) [latex]D: x eq 1\ left(frac{8}{5} ight)^{frac{2x + 1}{x-1}} = left(frac{125}{512} ight)^{3 - x} = left(frac{5}{8} ight)^{3(3 - x)} = left(frac{8}{5} ight)^{3(x - 3)}\ frac{2x + 1}{x-1} = 3(x - 3)\ 2x + 1 = 3(x - 3)(x - 1)\ 2x + 1 = 3x^2 - 12x + 9\ 3x^2 - 14x + 8 = 0\ Delta = 196 - 96 = 10^2\ x_1 = frac{14 - 10}{6} = frac{2}{3}\ x_2 = frac{14 + 10}{6} = 4[/latex]

Zad 5.7, 5.8, 5.9 oprócz zaznaczonych podpunktów

Zad 5.7, 5.8, 5.9 oprócz zaznaczonych podpunktów...