Question

Să de rezolve ecuațiile exponentiale. exercițiul E4

Answer 1

   
[tex]\displaystyle\\ E4.a) \\ 3^{x+1}+3^x \ 108 \\ 3^x\times 3^1+3^x \ 108 \\ 3^x(3+1)=108\\ 4\times 3^x = 108\\\\ 3^x = \frac{108}{4}\\\\ 3^x = 27 \\ 3^x=3^3\\ \boxed{x=3}\\\\ E4.b)\\ 3^{2x+1}+3^{2x}-3^{2x-1}=297\\ 3^{2x-1+2}+3^{2x-1+1}-3^{2x-1}=297\\ 3^{2x-1}\cdot 3^{2}+3^{2x-1}\cdot3^{1}-3^{2x-1}=297\\ 3^{2x-1}(3^{2}+3^{1}-1)=297\\ 3^{2x-1} \cdot 11 = 297\\ \\ 3^{2x-1} = \frac{297}{11} \\ \\ 3^{2x-1} =27\\ 3^{2x-1} =3^3\\ 2x+1 = 3\\ 2x = 2\\\\ \boxed{x =1} [/tex]


[tex]\displaystyle\\ E4.c)\\ 2^{x-2}+2^{x-3}+2^{x-4}=448\\ 2^{x-4+2}+2^{x-4+1}+2^{x-4}=448\\ 2^{x-4}\cdot 2^{2}+2^{x-4}\cdot 2^{1}+2^{x-4}=448\\ 2^{x-4}(2^{2}+2^{1}+1)=448\\ 2^{x-4}\cdot 7=448\\\\ 2^{x-4} = \frac{448}{7} \\\\ 2^{x-4} = 64\\ 2^{x-4} = 2^6\\ x-4=6\\ x=6+4\\ \boxed{x=10}[/tex]