[tex]a) {3}^{x} + {3}^{x - 1} = 4[/tex]
[tex] {3}^{x - 1} (3 + 1) = 4[/tex]
[tex] {3}^{x - 1} \times 4= 4 \: | \div 4[/tex]
[tex] {3}^{x - 1} = 1[/tex]
[tex] = > x = 1[/tex]
[tex] {3}^{1 - 1} = {3}^{0} = 1[/tex]
[tex]b) {3}^{x} + {3}^{x - 1} + {3}^{x - 2} + {3}^{x - 3} = 40[/tex]
[tex] {3}^{x - 3} ( {3}^{3} + {3}^{2} + 3 + 1) = 40[/tex]
[tex] {3}^{x - 3} (27 + 9 + 3 + 1) = 40[/tex]
[tex] {3}^{x - 3} \times 40 = 40 \: | \div 40[/tex]
[tex] {3}^{x - 3} = 1[/tex]
[tex] = > x = 3[/tex]
[tex] {3}^{3 - 3} = {3}^{0} = 1[/tex]
[tex]c)5 \times {3}^{x} - 2 \times {3}^{x} = 81[/tex]
[tex] {3}^{x} (5 - 2) = {3}^{4} [/tex]
[tex] {3}^{x} \times 3 = {3}^{4} [/tex]
[tex] {3}^{x + 1} = {3}^{4} [/tex]
[tex]x + 1 = 4[/tex]
[tex]x = 4 - 1[/tex]
[tex]x = 3[/tex]
[tex]d) {3}^{2x + 1} - {3}^{2x} - {3}^{2x - 1} - {3}^{2x - 2} - {3}^{2x - 3} = 123[/tex]
[tex] {3}^{2x - 3} ( {3}^{4} - {3}^{3} - {3}^{2} - 3 - 1) = 123[/tex]
[tex] {3}^{2x - 3} (81 - 27 - 9 - 3 - 1) = 123[/tex]
[tex] {3}^{2x - 3} \times 41 = 123[/tex]
[tex] {3}^{2x - 3} = 123 \div 41[/tex]
[tex] {3}^{2x - 3} = 3[/tex]
[tex]2x - 3 = 1[/tex]
[tex]2x = 1 + 3[/tex]
[tex]2x = 4 \: | \div 2[/tex]
[tex]x = 2[/tex]
