Question

1) Уравнение log_0.2 числа (5-6х)+2=0
2) неравенство log_1/3 log_4 числа (х^2 -5)>0

Answer 1

$1) log_{0,2}(5-6x)+2=0; \\ log_{0,2}(5-6x)=-2; \\ 0,2^{-2}=5-6x; \\ 5-6x=25; \\ 6x=5-25; \\ 6x=-20; \\ x=- \frac{20}{6}=- \frac{10}{3}=-3 \frac{1}{3}. \\ 5-6x\ \textgreater \ 0; \\ -6x\ \textgreater \ -5; \\ x\ \textless \ \frac{5}{6}.$
Ответ: -3 1/3.

$2) log_{1/3} log_{4}(x^2-5)\ \textgreater \ 0; \\ log_{1/3} log_{4}(x^2-5)\ \textgreater \ log_{1/3}1; \\ \left \{ {{ log_{4}(x^2-5)\ \textless \ 1, } \atop { log_{4}(x^2-5)\ \textgreater \ 0; }} \right. \\ \left \{ {{ log_{4}(x^2-5)\ \textless \ log_{4}4, } \atop { log_{4}(x^2-5)\ \textgreater \ log_{4}1; }} \right. \\ \left \{ {{x^2-5\ \textless \ 4,} \atop {x^2-5\ \textgreater \ 1;}} \right. \\ \left \{ {{x^2-9\ \textless \ 0,} \atop {x^2-6\ \textgreater \ 0;}} \right. \\ \left \{ {{(x-3)(x+3)\ \textless \ 0,} \atop {(x- \sqrt{6})(x+ \sqrt{6})\ \textgreater \ 0;}} \right.$
x∈(-3;3)
x∈(-∞;-√6)∪(√6;+∞).
Общее решение (см. на рисунке):
(-3;-√6)∪(√6;3).
Ответ: (-3;-√6)∪(√6;3).