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 lt; 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.