Question

va rog duin suflet ajutaima ! ! 

Answer 1

Regula este aceeasi pt toate exercitiile: identifici puterea cea mai mica si dai factor comun acea putere:
a) $3^{x+1} + 3^{x} =108$
$3^{x} *3+ 3^{x} =108$
$3^{x} *(3+ 1) =108$
$3^{x} *4 =4*27$ Impartim ambii membri la 4 si ramane:
$3^{x} =27$, adica
$3^{x} =3^{3}$, de unde x=3

b) $3^{2x+1} + 3^{2x} - 3^{2x-1} =297$
$3^{2x-1}*( 3^{2} + 3 - 1) =11*27$
$3^{2x-1}*11 =11*27$
$3^{2x-1} = 3^{3}$, deci
2x-1=3
2x=4
x=2

c) $2^{x-3} + 2^{x-2} + 2^{x-1} =448$
$2^{x-3}*(1 + 2 + 2^{2} ) =7*64$
$2^{x-3}*7 =7*64$
$2^{x-3} = 2^{6}$, deci
x-3=6
x=9

d) $3^{x+2} + 3^{x+1} + 3^{x} + 3^{x-1} + 3^{x-2} + 3^{x-3} =1092$
$3^{x-3} *( 3^{5} + 3^{4} + 3^{3} + 3^{22} + 3 +1)=1092$

Folosim rezultatul cunoscut:
$1+x+ x^{2} + x^{3} + x^{4} + ... + x^{n} = \frac{x^{n+1} - 1}{x-1}$

In cazul nostru, n=5, deci avem:
$3^{x-3} * \frac{3^{6} - 1}{3-1} =1092$
$3^{x-3} * \frac{729 - 1}{2} =1092$
$3^{x-3} * \frac{728}{2} =1092$
$3^{x-3} *364 =3*364$
$3^{x-3} =3$, deci
x-3=1
x=4

e) $5^{x-1} + 5^{x-2} + 5^{x-3} =775$
$5^{x-3}*(1 + 5 + 5^{2} ) =31*25$
$5^{x-3}*31 =31*25$
$5^{x-3} =25$
$5^{x-3} = 5^{2}$, deci
x-3=2
x=5

f) Nu stiu daca am inteles bine:
$3* 7^{x} +3* 7^{x+1} + 7^{x+2} =219$
$7^{x} *(3 + 3* 7 + 7^{2} ) =3*73$
$7^{x} *73 =3*73$
$7^{x} =3$
logaritmi?!?!?!.... poate este gresit ex...