Réponse :
Explications étape par étape
Ln a + Ln b = ln (ab)
ln a - Ln b = ln (a/b)
Bonsoir
Résoudre les équations :
ln(x + 2) - ln(5 - 2x) = Ln(x + 3)
ln (x + 2)/(5 - 2x) = Ln (x + 3)
(x + 2)/(5 - 2x) = x + 3
Avec : 5 - 2x # 0
2x # 5
x # 5/2
• x + 2 > 0
x > -2
• 5 - 2x > 0
5 > 2x
x < 5/2
• x + 3 > 0
x > -3
x + 2 = (x + 3)(5 - 2x)
x + 2 = 5x - 2x^2 + 15 - 6x
2x^2 + 2x + 2 - 15 = 0
2x^2 + 2x - 13 = 0
[tex]\Delta = 2^{2} - 4 * 2 * (-13) = 4 + 104 = 108[/tex]
[tex]\sqrt{\Delta} ~ 10,4
x1 = (-2 - 10,4)/(2 * 2) = -12,4/4 = -3,1
x2 = (-2 + 10,4)/4 = 8,4/4 = 2,1
Seule réponse possible 2,1
ln(x^2 - 1) << ln(4x - 1) - 2ln 2
x^2 - 1 > 0
(x - 1)(x + 1) > 0
x.........|-inf..........(-1)...........1...........+inf
x-1......|.........(-)...........(-).....o....(+)........
x+1.....|.........(-).....o....(+)...........(+).......
Ineq...|.........(+)....o.....(-)...o....(+).......
x € ]-inf ; -1[ U ]1 ; +inf[
4x - 1 > 0
4x > 1
x > 1/4
2ln 2 = Ln 2^2
Ln(x^2 - 1) << Ln (4x - 1)/(2^2)
(x - 1)(x + 1) << (4x - 1)/4
4(x - 1)(x + 1) << 4x - 1
4(x^2 - 1) - 4x + 1 << 0
4x^2 - 4 - 4x + 1 << 0
4x^2 - 4x - 3 << 0
[tex]\Delta = (-4)^{2} - 4 * 4 * (-3) = 16 + 48 = 64[/tex]
[tex]\sqrt{\Delta} = 8[/tex]
x1 = (4 - 8)/(2 * 4) = (-4/8) = (-1/2)
x2 = (4 + 8)/8 = 12/8 = 3/2
x.......|-inf...........(-1/2).........3/2..........+inf
Ineq.|.......(+)........o.....(-)......o.....(+)........
x € [-1/2 ; 3/2]
(7x - 3) ln(x + 3) > 0
7x - 3 > 0
7x > 3
x > 3/7
Ln (x + 3) > 0
x + 3 > 0
x > -3
2(Ln x)^2 + Ln x - 6 = 0
On remplace ln x par X :
2X^2 + X - 6 = 0
[tex]\Delta = 1^{2} - 4 * 2 * (-6) = 1 + 48 = 49[/tex]
[tex]\sqrt{\Delta} = 7[/tex]
X1 = (-1 - 7)/(2 * 2) = -8/4 = (-2)
X2 = (-1 + 7)/4 = 6/4 = 3/2
Ln x = -2
x = e^(-2)
Ou
Ln x = 3/2
x = e^(3/2)
