Решите уравнения: а) log7 (3x-5) - log7 (9-2x) = 1 б) 4 - lg^2 x = 3lg x

а)

1) ОДЗ:

{3x - 5 > 0;

{9 - 2x > 0; {x > 5/3;

{x < 9/2; x ∈ (5/3; 9/2).

2) Разность логарифмов:

log7 (3x - 5) - log7 (9 - 2x) = 1; log7 ((3x - 5) / (9 - 2x)) = log7 (7); (3x - 5) / (9 - 2x) = 7; 3x - 5 = 7 (9 - 2x); 3x - 5 = 63 - 14x; 3x + 14x = 63 + 5; 17x = 68; x = 68 : 17; x = 4.

б)

4 - lg^2x = 3lgx; lg^2x + 3lgx - 4 = 0; D = 3^2 + 4 * 4 = 9 + 16 = 25; lgx = (-3 ± √25) / 2 = (-3 ± 5) / 2;

1) lgx = (-3 - 5) / 2 = - 8/2 = - 4;

lgx = - 4; x = 10^ (-4); x = 0,0001;

2) lgx = (-3 + 5) / 2 = 2/2 = 1;

lgx = 1; x = 10^1; x = 10.

Ответ:

а) 4; б) 0,0001; 10.