1/2log2 (x-4) + 1/2log2 (2x-1) = log2 3

1. Область допустимых значений:

1/2 * log2 (x - 4) + 1/2 * log2 (2x - 1) = log2 (3); {x - 4 > 0;

{2x - 1 > 0; {x > 4;

{2x > 1; {x > 4;

{x > 1/2; x ∈ (4; ∞).

2. Сумма логарифмов:

1/2 (log2 (x - 4) + log2 (2x - 1)) = log2 (3); 1/2 * log2 ((x - 4) (2x - 1)) = log2 (3); log2 ((x - 4) (2x - 1)) = 2log2 (3); log2 ((x - 4) (2x - 1)) = log2 (3^2); (x - 4) (2x - 1) = 9; 2x^2 - 9x + 4 - 9 = 0; 2x^2 - 9x - 5 = 0; D = 9^2 + 4 * 2 * 5 = 81 + 40 = 121; x = (9 ± √121) / 4 = (9 ± 11) / 4; x = (9 - 11) / 4 = - 2/4 = - 1/2 - не подходит; x = (9 + 11) / 4 = 20/4 = 5.

Ответ: 5.