Question

a) 3x2^x+2-5x2^x+1-2^x=2
b)5^x+2-2x5^-12x5^x=375
c)2^x . 3^x+1=108
d)3^x+1 . 5^x=675
e) 3^x+2 . 7^x - 3^x . 7^x+1 =2940
DAU COROANA :P

Answer 1

a)
[tex]3\cdot2^{x+2}-5\cdot2^{x+1}-2^x=2\\ 3\cdot2^x\cdot2^2-5\cdot2^x\cdot2^1-2^x=2\\ 2^x(3\cdot2^2-5\cdot2^1-1)=2\\ 2^x=2^1 \rightarrow \boxed{x=1}[/tex]

b)
[tex]5^{x+2}-2\cdot5^{-12}\cdot5^x=375\\ 5^x\cdot5^2-5^x\cdot2\cdot5^{-12}=375\\ 5^x(5^2-2\cdot5^{-12})=375\\ 5^x\cdot25=375\\ 5^x=15\\ \text{Stop pentru gimnaziu}\\ x=log_515[/tex]

c)
[tex]2^x\cdot3^{x+1}=108\\ \text{Desompunerea in factori primi:}\\ 108=2^2\cdot3^3\rightarrow \boxed{x=2}[/tex]

d)
[tex]3^{x+1}\cdot5^x=675\\ \text{Descompunerea in factori primi:}\\ 675=3^3\cdot5^2\rightarrow \boxed{x=2}[/tex]

e)
[tex]3^{x+2}\cdot7^{x}-3^x\cdot7^{x+1}=2940\\ 3^x\cdot7^x\cdot3^2-3^x\cdot7^x\cdot7^1=2940\\ 3^x\cdot7^x(3^2-7^1)=2940\\ 3^x\cdot7^x\cdot2=2940\\ 3^x\cdot7^x=1470\\ 21^x=1470\\ \text{Pentru gimnaziu lucrurile se opresc aici stim despre x ca este}\\ \text{cuprins intre 2 si 3}\\ x=log_{21}1470[/tex]