Lg (x+√3) + lg (x-√3) = 0 log₂ (x-5) + log₂ (x+2) = 3 log₃x+logx3=5/2

1. Сумма логарифмов:

lg (x + √3) + lg (x - √3) = 0; lg ((x + √3) (x - √3)) = 0; lg (x^2 - 3) = 0; x^2 - 3 = 1; x^2 = 4; x1 = - 2 - не входит в область допустимых значений; x2 = 2.

2. Сумма логарифмов:

log2 (x - 5) + log2 (x + 2) = 3; log2 ((x - 5) (x + 2)) = 3; (x - 5) (x + 2) = 2^3; x^2 - 3x - 10 = 8; x^2 - 3x - 18 = 0; D = 3^2 + 4 * 18 = 9 + 72 = 81 = 9^2; x = (3 ± 9) / 2; x1 = (3 - 9) / 2 = - 6/2 = - 3 - не входит в область допустимых значений; x2 = (3 + 9) / 2 = 12/2 = 6.

3. Переход к другому основанию:

log3 (x) + logx (3) = 5/2; log3 (x) + 1/log3 (x) = 5/2; (log3 (x)) ^2 - 5/2 * log3 (x) + 1 = 0; 2 (log3 (x)) ^2 - 5log3 (x) + 2 = 0; D = 5^2 - 4 * 2 * 2 = 25 - 16 = 9 = 3^2; log3 (x) = (5 ± 3) / 4;

a)

log3 (x) = (5 - 3) / 4 = 2/4 = 1/2; x = 3^ (1/2) = √3;

b)

log3 (x) = (5 + 3) / 4 = 8/4 = 2; x = 3^2 = 9.

Ответ: 1) 2; 2) 6; 3) √3; 9.