10 вычислить log2 (4x-3) + log1/8 (125) = log0.5 (x) + log4 (0.04)

1. ОДЗ:

{4 х - 3 > 0;

{х > 0; {4 х > 3;

{х > 0; {х > 3/4;

{х > 0; х ∈ (3/4; ∞).

2. Приведем к основанию 2:

log2 (4 х - 3) + log (1/8) (125) = log0,5 (х) + log4 (0,04); log2 (4 х - 3) + log2 (125) / log2 (1/8) = log2 (х) / log2 (0,5) + log2 (0,04) / log2 (4); log2 (4 х - 3) + 3log2 (5) / (-3) = log2 (х) / (-1) - 2log2 (5) / 2; log2 (4 х - 3) - log2 (5) = - log2 (х) - log2 (5); log2 (4 х - 3) + log2 (х) = 0; log2 (х (4 х - 3)) = 0; х (4 х - 3) = 1; 4 х² - 3 х - 1 = 0; D = 3² + 4 * 4 = 25; х = (3 ± 5) / 2; х1 = (3 - 5) / 8 = - 1/4 ∉ ОДЗ; х2 = (3 + 5) / 8 = 1.

Ответ: 1.