1) lg (6*5^x-25*20^x) - lg25=x 2) lg^2 (x+1) = lg (x+1) lg (x-1) + 2lg^2 (x-1)

1.

lg (6 * 5^x - 25 * 20^x) - lg25 = x;

lg ((6 * 5^x - 25 * 4^x * 5^x) / 25) = x;

(6 * 5^x - 25 * 4^x * 5^x) / 25 = 10^x;

6 * 5^x - 25 * 4^x * 5^x = 25 * 10^x;

6 * 5^x - 25 * 4^x * 5^x - 25 * 2^x * 5^x = 0;

5^x (6 - 25 * 4^x - 25 * 2^x) = 0;

6 - 25 * 2^ (2x) - 25 * 2^x = 0;

25 * 2^ (2x) + 25 * 2^x - 6 = 0;

D = 25^2 + 4 * 25 * 6 = 625 + 600 = 1225 = 35^2;

2^x = (-25 ± 35) / 50;

1) 2^x = (-25 - 35) / 50 < 0 - нет решения;

2) 2^x = (-25 + 35) / 50 = 10/50 = 1/5;

x = log2 (1/5) = - log2 (5).

2. Пусть:

lg (x + 1) = u;

lg (x - 1) = v.

Тогда:

log^2 (x + 1) = lg (x + 1) lg (x - 1) + 2lg^2 (x - 1);

u^2 = uv + 2v^2;

u^2 - uv - 2v^2 = 0;

(u/v) ^2 - (u/v) - 2 = 0;

D = 1^2 + 4 * 2 = 9;

1) u/v = (1 - 3) / 2 = - 1;

lg (x + 1) / lg (x - 1) = - 1;

lg (x + 1) = - lg (x - 1);

lg (x + 1) + lg (x - 1) = 0;

lg ((x + 1) (x - 1)) = 0;

x^2 - 1 = 1;

x^2 = 2;

[x = - √2 ∉ ОДЗ;

[x = √2 - корень.

2) u/v = (1 + 3) / 2 = 2;

lg (x + 1) / lg (x - 1) = 2;

lg (x + 1) = 2lg (x - 1);

lg (x + 1) = lg (x - 1) ^2;

x + 1 = x^2 - 2x + 1;

x^2 - 3x = 0;

x (x - 3) = 0;

[x = 0 ∉ ОДЗ;

[x = 3 - корень.