Zad 5.7, 5.8, 5.9 oprócz zaznaczonych podpunktów

zadanie 5.7 e) [latex]100^x = 625*4^x\ (25*4)^x = 5^4*2^{2x}\ 5^{2x}*2^{2x} = 5^4*2^{2x} |:2^{2x}\ 5^{2x} = 5^4\ x = 2[/latex]   f) [latex]5*2^{sqrt{x}} - 3*2^{sqrt{x} - 1} = 56\ 2^{sqrt{x} - 1}(5*2 - 3) = 2^{sqrt{x} - 1}*7 = 56\ 2^{sqrt{x} - 1} = frac{56}{7} = 8 = 2^3\ sqrt{x} = 3 + 1 = 4\ x = 16[/latex]   g) [latex]2*5^{sqrt{x}} + 3*5^{sqrt{x} - 1} = 325\ 5^{sqrt{x} - 1}(2*5 + 3) = 5^{sqrt{x} - 1}*13 = 325\ 5^{sqrt{x} - 1} = frac{325}{13} = 25 = 5^2\ sqrt{x} = 2 + 1 = 3\ x = 9[/latex]   h) [latex]2^{3sqrt{x}} + 2^{3sqrt{x} - 1} - 2^{3sqrt{x} - 2} = 10\ 2^{3sqrt{x} - 2}(2^2 + 2 - 1) = 2^{3sqrt{x} - 2}*5 = 10\ 2^{3sqrt{x} - 2} = frac{10}{5} = 2 = 2^1\ sqrt{x} = frac{1 + 2}{3} = 1\ x = 1[/latex]     zadanie 5.8 b) [latex]7^{2x} - 3*7^x - 28 = 0\ t = 7^x > 0\ t^2 - 3t - 28 = 0\ (t - 7)(t + 4) = 0\ t = 7^x = 7 Rightarrow x = 1[/latex]   c) [latex]5*2^{x + 1} - 2^{2x} = 16\ 2^{2x} - 10*2^x + 16 = 0\ (2^x - 8)(2^x - 2) = 0\ (2^x - 2^3)(2^x - 2^1) = 0\ x = 3 vee x = 1[/latex]   d) [latex]3^{2x} - 10*3^{x+1} + 81 = 0\ 3^{2x} - 30*3^{x} + 81 = 0\ t = 3^x > 0\ t^2 - 30t + 81 = 0\ (t - 27)(t - 3) = 0\ t_1 = 3^{x_1} = 3 Rightarrow x_1 = 1\ t_2 = 3^{x_2} = 27 Rightarrow x_2 = 3[/latex]   e) [latex]5^{2x - 3} = 2*5^{x - 2} + 3\ frac{1}{125}5^{2x} - frac{2}{25}5^x - 3 = 0 |*125\ t = 5^x > 0\ t^2 - 10t - 375 = 0\ (t - 25)(t + 15) = 0\ t = 5^x = 25 = 5^2 Rightarrow x = 2[/latex]   f) [latex]3^{2x - 2} - 12 = - 3^{x - 1}\ (3^{x - 1})^2 + 3^{x - 1} - 12 = 0\ (3^{x - 1} + 4)(3^{x - 1} - 3) = 0\ 3^{x - 1} = 3 = 3^1\ x = 2[/latex]   g) [latex]2*4^{2x} - 17*4^x + 8 = 0\ t = 4^x > 0\ 2t^2 - 17t + 8 = 0\ Delta = 289 - 64 = 225 = 15^4\ t_1 = 4^{x_1} = frac{17 - 15}{4} = frac{1}{2} = 2^{-1} = 4^{-frac{1}{2}} Rightarrow x = -frac{1}{2}\ t_2 = 4^{x_2} = frac{17 + 15}{4} = 8 = 2^{3} = 4^{frac{3}{2}} Rightarrow x = frac{3}{2}[/latex]   h) [latex]2frac{1}{4}*4^{2x} - frac{1}{2}*4^{4x} = 1\ 2*4^{4x} - 9*4^{2x} + 4 = 0\ t = 4^{2x} > 0\ 2t^2 - 9t + 4 = 0\ Delta = 81 - 32 = 49 = 7^2 t_1 = 4^{2x_1} = frac{9 + 7}{4} = 4 Rightarrow x_1 = frac{1}{2}\ t_2 = 4^{2x_2} = frac{9 - 7}{4} = frac{1}{2} = 4^{-frac{1}{2}} Rightarrow x_2 = -frac{1}{4}[/latex]   i) [latex]5^{2x - 1} + 5^{x + 1} = 250\ frac{1}{5}5^{2x} + 5*5^x - 250 = 0\ 5^{2x} + 25*5^x - 1250 = 0\ (5^x - 25)(5^x + 50) = 0\ 5^x = 25 = 5^2\ x = 2[/latex]     zadanie 5.9 a) [latex]12*2^x - 4^x = 32\ 2^{2x} - 12*2^x + 32\ (2^x - 8)(2^x - 4) = 0\ (2^x - 2^3)(2^x - 2^2) = 0\ x = 3 vee x = 2[/latex]   b) [latex]6*2^x - 4^x = 8\ 2^{2x} - 6*2^x + 8 = 0\ (2^x - 4)(2^x - 2) = 0\ (2^x - 2^2)(2^x - 2^1) = 0\ x = 2 vee x = 1[/latex]   c) [latex]4^{x - 2} - 17*2^{x - 4} + 1 = 0\ 2^{2x - 4} - 17*2^{x - 4} + 1 = 0 |*2^4\ 2^{2x} - 17*2^x + 16\ (2^x - 16)(2^x - 1) = 0\ (2^x - 2^4)(2^x - 2^0) = 0\ x = 4 vee x = 0[/latex]   d) [latex]3*9^{x - 2} - 28*3^{x - 3} + 1 = 0\ 3^{2x - 3} - 28*3^{x - 3} + 1 = 0 |*3^3\ 3^{2x} - 28*3^x + 27 = 0\ (3^x - 27)(3^x - 1) = 0\ (3^x - 3^3)(3^x - 3^0) = 0\ x = 3 vee x = 0[/latex]   f) [latex]36^x + 8*6^x - 84 = 0\ 6^{2x} + 8*6^x - 84 = 0\ Delta = 64 + 4*84 = 16(4 + 21) = (4*5)^2 = 20^2\ 6^x > 0\ 6^{x} = frac{- 8 + 20}{2} = 6 Rightarrow x = 1[/latex]

Zad 5.10, 5.11, 5.12 oprócz zaznaczonych podpunktów  

Zad 5.10, 5.11, 5.12 oprócz zaznaczonych podpunktów  ...