[tex] E4)a) {3}^{x + 1} + {3}^{x} = 108[/tex]
[tex] {3}^{x} (3 + 1) = 108[/tex]
[tex] {3}^{x} \times 4 = 108 \: | \div 4[/tex]
[tex] {3}^{x} = 27 = > x = 3[/tex]
[tex]b) {3}^{2x + 1} + {3}^{2x} - {3}^{2x - 1} = 297[/tex]
[tex] {3}^{2x - 1} ( {3}^{2} + 3 - 1) = 297[/tex]
[tex] {3}^{2x - 1} (9 + 3 - 1) = 297[/tex]
[tex] {3}^{2x - 1} \times 11 = 297 \: | \div 11[/tex]
[tex] {3}^{2x - 1} = 27[/tex]
[tex] {3}^{2x - 1} = {3}^{3} [/tex]
[tex]2x - 1 = 3[/tex]
[tex]2x = 3 + 1[/tex]
[tex]2x = 4 \: | \div 2[/tex]
[tex]x = 2[/tex]
[tex]E5)a) {4}^{x} + {2}^{x + 1} = 80[/tex]
[tex] { ({2}^{x} )}^{2} + {2}^{x} \times 2 = 80[/tex]
[tex] {2}^{x} = t \: ,t > 0[/tex]
[tex] {t}^{2} + 2t = 80[/tex]
[tex] {t}^{2} + 2t - 80 = 0[/tex]
[tex]a = 1[/tex]
[tex]b = 2[/tex]
[tex]c = - 80[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {2}^{2} - 4 \times 1 \times ( - 80)[/tex]
[tex]\Delta = 4 + 320[/tex]
[tex]\Delta = 324[/tex]
[tex]t_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a}[/tex]
[tex]t_{1,2}=\frac{-2\pm\sqrt{324}}{2 \times 1}[/tex]
[tex]t_{1,2}=\frac{-2\pm18}{2}[/tex]
[tex]t_{1}=\frac{-2 + 18}{2} = \frac{16}{2} = 8[/tex]
[tex]t_{2}=\frac{-2 - 18}{2} = - \frac{20}{2} = - 10 \: < 0 = > \: nu \: convine[/tex]
[tex] {2}^{x} = t = >{2}^{x} = 8 = > x = 3[/tex]
[tex]b) {9}^{x} - 10 \times {3}^{x} + 9 = 0[/tex]
[tex] ({3}^{x} )^{2}- 10 \times {3}^{x} + 9 = 0[/tex]
[tex] {3}^{x} = t \: ,t > 0[/tex]
[tex] {t}^{2} - 10t + 9 = 0[/tex]
[tex]a = 1[/tex]
[tex]b = - 10[/tex]
[tex]c = 9[/tex]
[tex]\Delta = {b}^{2} - 4ac[/tex]
[tex]\Delta = {( - 10)}^{2} - 4 \times 1 \times 9[/tex]
[tex]\Delta = 100 - 36[/tex]
[tex]\Delta = 64[/tex]
[tex]t_{1,2}=\frac{-b\pm\sqrt{\Delta}}{2a}
[/tex]
[tex]t_{1,2}=\frac{-( - 10)\pm\sqrt{64}}{2 \times 1}
[/tex]
[tex]t_{1,2}=\frac{10\pm8}{2}
[/tex]
[tex]t_{1}=\frac{10 + 8}{2} = \frac{18}{2} = 9
[/tex]
[tex]t_{2}=\frac{10 - 8}{2} = \frac{2}{2} = 1
[/tex]
[tex] {3}^{x} = t[/tex]
[tex] = > {3}^{x} = 9 = > x = 2[/tex]
[tex] = > {3}^{x} = 1 = > x = 0[/tex]
