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Консультация № 197533
05.01.2020, 22:30
0.00 руб.
0
4
1
Здравствуйте! У меня возникли сложности с таким вопросом:
lg(x^4+2x^3-2x^2-2x+1)=lg(2x^3-2x^2+2)
lg(x^4+2x^3-2x^2-2x+1)=lg(2x^3-2x^2+2)
Обсуждение
05.01.2020, 22:38
общий
В примере была допущено ошибка при наборе, вот правильный пример
lg(x^4+2x^3-2x^2-2x+1)=lg(2x^3-2x+2)
lg(x^4+2x^3-2x^2-2x+1)=lg(2x^3-2x+2)
06.01.2020, 16:46
общий
это ответ
Здравствуйте, Mari!
Обозначим y1(x) = x4 + 2x3 - 2x2 - 2x + 1
y2(x) = 2x3 - 2x + 2
Поскольку логарифмы функций y1(x) и y2(x) равны, значит можно заменить уравнение более простым:
y1(x) = y2(x)
Однако запомним, что из конечного результата надо будет отбраковать варианты, при которых y1(x)[$8804$]0 либо y2(x)[$8804$]0 , ибо выражение под логарифмом должно быть больше нуля!
Из y1(x) = y2(x) вытекает : x4 - 2x2 - 1 = 0
Заменим x2 = b
b2 - 2b - 1 = 0
Выделим полный квадрат:
b2 - 2b +1 - 2 = (b-1)2 - 2 = 0
b-1 = [$177$][$8730$]2
x2 = 1 [$177$][$8730$]2
x = [$177$][$8730$](1[$177$][$8730$]2)
Из полученных 4х вариантов решений отбрасываем сначала x1 = +[$8730$](1- [$8730$]2) и x2 = -[$8730$](1- [$8730$]2) , поскольку корень из отрицательного числа находится вне множества вещественных чисел.
Из 2х оставшихся вариантов x3 = +[$8730$](1+[$8730$]2) и x4 = -[$8730$](1+[$8730$]2) бракуем отрицательный x4 , поскольку y1(x4) = -2,395 недопустимо для логарифмирования в области вещественных чисел.
Ответ x = x3 = [$8730$](1+[$8730$]2) = 1,554
Проверяем : y1(1,554) = 6,395 , y2(1,554) = 6,395
lg(6,395) = 0,806 - Проверка успешна!
Обозначим y1(x) = x4 + 2x3 - 2x2 - 2x + 1
y2(x) = 2x3 - 2x + 2
Поскольку логарифмы функций y1(x) и y2(x) равны, значит можно заменить уравнение более простым:
y1(x) = y2(x)
Однако запомним, что из конечного результата надо будет отбраковать варианты, при которых y1(x)[$8804$]0 либо y2(x)[$8804$]0 , ибо выражение под логарифмом должно быть больше нуля!
Из y1(x) = y2(x) вытекает : x4 - 2x2 - 1 = 0
Заменим x2 = b
b2 - 2b - 1 = 0
Выделим полный квадрат:
b2 - 2b +1 - 2 = (b-1)2 - 2 = 0
b-1 = [$177$][$8730$]2
x2 = 1 [$177$][$8730$]2
x = [$177$][$8730$](1[$177$][$8730$]2)
Из полученных 4х вариантов решений отбрасываем сначала x1 = +[$8730$](1- [$8730$]2) и x2 = -[$8730$](1- [$8730$]2) , поскольку корень из отрицательного числа находится вне множества вещественных чисел.
Из 2х оставшихся вариантов x3 = +[$8730$](1+[$8730$]2) и x4 = -[$8730$](1+[$8730$]2) бракуем отрицательный x4 , поскольку y1(x4) = -2,395 недопустимо для логарифмирования в области вещественных чисел.
Ответ x = x3 = [$8730$](1+[$8730$]2) = 1,554
Проверяем : y1(1,554) = 6,395 , y2(1,554) = 6,395
lg(6,395) = 0,806 - Проверка успешна!
5
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