a)
[tex] {3}^{x + 1} + {3}^{x} = 108 \\ 3 {}^{x} \times 3 + {3}^{x} = 108[/tex]
[tex]Notam \: \: {3}^{x} = t[/tex]
[tex]t + 3t = 108 \\ 4t = 108 \\ t = 108 \div 4 \\ t = 27[/tex]
[tex] {3}^{x} = 27 \\ {3}^{x} = {3}^{3} [/tex]
→ (Injectivitate) x=3
b)
[tex]3 {}^{2x + 1} + {3}^{2x} - {3}^{2x - 1} = 297 \\ {3}^{2x} \times 3 + 3 {}^{2x} - 3 {}^{2x} \times 3 {}^{ - 1} = 297 \\ \\ Notam \: \: {3}^{2x} = a \\ 3a + a - \frac{1}{3} a = 297 \\ {}^{3)} 4a - \frac{a}{3} = 297 \\ \frac{12a - a}{3} = 297 \\ 11a = 297 \div 3 \\ 11a = 99 \\ a = 99 \div 11 \\ a = 9[/tex]
[tex] {3}^{2x} = 9 \\ {3}^{2x} = 3 {}^{2} [/tex]
→ (injectivitate) 2x=2 |:2
x=2
