Корень из log по основанию 3 числа x^9 - 4log по основанию 9 числа корень из 3x = 1

1. Преобразуем:

√log3 (х^9) - 4log9 (√ (3 х)) = 1; √ (9 * log3 (х)) - 4 * log3 (√ (3 х)) / log3 (9) = 1; 3√ (log3 (х)) - 4 * log3 (√ (3 х)) / 2 = 1; 3√ (log3 (х)) - 2 * log3 (√ (3 х)) = 1; 3√ (log3 (х)) - 2 * 1/2 * log3 (3 х) = 1; 3√ (log3 (х)) - log3 (3 х) = 1; 3√ (log3 (х)) - log3 (3) - log3 (х) = 1; 3√ (log3 (х)) - 1 - log3 (х) - 1 = 0; 3√ (log3 (х)) - log3 (х) - 2 = 0; log3 (х) - 3√ (log3 (х)) + 2 = 0.

2. Обозначим:

√ (log3 (х)) = t.

Тогда:

t^2 - 3t + 2 = 0; D = 3^2 - 4 * 2 = 9 - 8 = 1; t = (3 ± 1) / 2;

1) t = (3 - 1) / 2 = 2/2 = 1;

√ (log3 (х)) = 1; log3 (х) = 1; х = 3;

2) t = (3 + 1) / 2 = 4/2 = 2;

√ (log3 (х)) = 2; log3 (х) = 2^2 = 4; х = 3^4 = 81.

Ответ: 3; 81.