1) lg (5-x) + lgx=lg4; 2) lg (x+1) + lg (x-1) = lg3; 3) ln (6-x) + lnx=ln5; 4) lgx+lg (x-3) = 1

1) lg (5 - x) + lgx = lg4.

Используя свойство логарифма произведения:

lg ((5-x) * х) = lg4.

(5 - x) * х = 4.

x2 - 5x + 4 = 0.

D = 25 - 4 * 1 * 4 = 9.

√D = 3.

Т. к. D > 0, то корней два.

x1 = (5 + 3) / 2 = 4.

x2 = (5 - 3) / 2 = 1.

2) ln (6 - x) + lnx = ln5.

ln ((6 - x) * x) = ln5.

(6 - x) * x = 5.

x2 - 6x + 5 = 0.

D=36 - 4 * 1 * 5 = 16.

√D = 4.

x1 = (6 + 4) / 2 = 5.

x2 = (6 - 4) / 2 = 1.

3) lg (x + 1) + lg (x - 1) = lg3.

lg ((x + 1) (x - 1)) = lg3.

(x + 1) (x - 1) = 3.

x2 - 1 = 3.

x2 = 4.

x1 = - 2. Не подходит т. к. - 2 - 3 = - 5, а lg (-5) не имеет смысла по определению.

x2 = 2.

4) lgx + lg (x - 3) = 1.

lg (x (x - 3)) = lg10.

x (x - 3) = 10.

x2 - 3x - 10 = 0.

D = 9 - 4 * 1 * 10 = 49.

x1 = (3 + 7) / 2 = 5.

x2 = (3 - 7) / 2 = - 2. Не подходит.