1)
2^(x^2-8x+18) = 8
2^(x^2-8x+18) = 2^3
x^2-8x+18 = 3
x^2-8x+15 = 0
D = 64 - 60 = 4
x1 = (8-4)/2 = 2
x2 = (8+4)/2 = 6
2)
49^x+5 7^x - 14 = 0
Обозначим 7^x = t > 0
t^2 + 5t - 14 = 0
D = 25 +56 = 81
t1 = (-5-9)/2 = -7 < 0 (t>0)
t2 = (-5+9)/2 = 2
7^x = 2
log (7) (7^x) = log (7) (2)
x = log (7) (2)
3)
log (4) (6x^2 + 18x + 16) = 2
log (4) (6x^2 + 18x + 16) = 2 log (4) (4)
log (4) (6x^2 + 18x + 16) = log (4) (16)
6x^2 + 18x + 16 = 16
2x(x+3) = 0
x1 = 0
x2 = -3
(Выражение в логарифме 6x^2 + 18x + 16 > 0 при x = 0 и x = -3)
4)
4 lg(x)^2 +lg(x^3) = 1
4 lg(x)^2 + 3lg(x) - 1 = 0
Обозначим: lg(x) = t
4 t^2 + 3 t - 1 = 0
D = 9 + 16 = 25
t1 = (-3-5)/8 = -1
t2 = (-3+5)/8 = 1/4
lg(x1) = t1
lg(x1) = -1
lg(x1) = - lg(10)
lg(x1) = lg(1/10)
x1 = 1/10
lg(x2) = 1/4
lg(x2) = (1/4)lg(10)
lg(x2) = lg(10^(1/4))
x2 = 10^(1/4)
x1 = 1/10
x2 = 10^(1/4)