решите систему логарифмических уравнений в) и е). Created by volochkos. algebra-ru

В)ОДЗ: х >0; y > 0{log₁/₃ x + log₁/₃ y = 2{log₁/₂ x - log₁/₂ y =4 ║ ∨{log₁/₃ x·y = 2{log₁/₂ x/y =4 ║ ∨{xy = (¹/₃)²{x/y =(¹/₂)⁴ ║ ∨{ x·y = ¹/₉{ x/y = ¹/₁₆ ║ ∨{x·y = ¹/₉{y =16x ║ ∨{y=16x{x·16x = ¹/₉ ║ ∨16x² = ¹/₉ 16x² · 9 = 1 144x² = 1144x² - 1 = 0(12x+1)(12x-1) = 012x+1 = 0 => 12x = -1 => x₁ = - 1/12 - не удовлетв. ОДЗ12x-1 = 0 => 12x = 1 => x₂ = 1/12 Найдём у при х = 1/12 подставив в у = 16х.у = 16 · 1/12 = 16/12 = 4/3 = Ответ: х = у = e)ОДЗ: х >0; y > 0{log₂ x + log₄ y = 4{log₄ x + log₂ y = 5 ║ ∨{log₂ x + 1/2 log₂ y = 4{1/2log₂ x + log₂ y = 5 ║ ∨{2log₂ x + log₂ y = 4 · 2{log₂ x + 2log₂ y = 5 · 2 ║ ∨{log₂ x² + log₂ y = 8{log₂ x + log₂ y² = 10 ║ ∨{log₂ x²y = 8{log₂ xy² = 10 ║ ∨{x²y = 2⁸{xy² = 2¹⁰Делим второе уравнение на первое:xy²/x²y = 2¹⁰/2⁸у/х= 2²у/х = 4у = 4хПодставим у = 4х в уравнение х²у = 2⁸ и найдем х:х² · 4х = 2⁸4х³ = 2⁸х³ = 2⁸ : 4 х³ = 2⁸ : 2²х³ = 2⁶х³ = 64х = ∛64 х = 4Найдём у = 4х при х = 4у = 4·4 = 16Ответ: х=4; у = 16 lg(xy)=3 и lgx*lgy=2 это система, не пойму как второе уравнение преобразоватьесли не трудно обьясните пожалуйстаНовые вопросы в АлгебраОцінити значення вразу 4b, якщо -3 <b<2Спростити вираз: -17m⁶n*(-0.5m*n¹³Реши уравнение можно с решением поподробнее. Скільки цілих чисел містить множина розв'язкiв нерівності: 4 - 9х ≤11? 2 -9≤ 22. Реши уравнение можно с решениемРеши уравнение можно с решениемРозв'язати нерiвнiсть: x+13 > -8x + 22 Разложи на множители можно пожалуйста поподробнее Разложи на множители можно только вариант ответаРазложи на множители можно с решениемПредыдущийСледующийВместе мы знаем большеЭтот сайт использует cookies Политика Cookies . 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решите систему логарифмических уравнений в) и е)...

volochkos @volochkos

July 2022 1 7 Report

решите систему логарифмических уравнений в) и е)

Answers & Comments

zinaidazina В)
ОДЗ: х >0; y > 0

{log₁/₃ x + log₁/₃ y = 2
{log₁/₂ x - log₁/₂ y =4
  ║
  ∨
{log₁/₃ x·y = 2
{log₁/₂ x/y =4
║
  ∨
{xy = (¹/₃)²
{x/y =(¹/₂)⁴
║
∨
{ x·y = ¹/₉
{ x/y = ¹/₁₆
║
∨
{x·y = ¹/₉
{y =16x
║
∨
{y=16x
{x·16x = ¹/₉
║
∨
16x² = ¹/₉
16x² · 9 = 1
144x² = 1
144x² - 1 = 0
(12x+1)(12x-1) = 0
12x+1 = 0 => 12x = -1 => x₁ = - 1/12 - не удовлетв. ОДЗ
12x-1 = 0 => 12x = 1 => x₂ = 1/12

Найдём у при х = 1/12 подставив в у = 16х.
у = 16 · 1/12 = 16/12 = 4/3 = $1 \frac{1}{3}$
Ответ: х = $\frac{1}{12};$ у = $1 \frac{1}{3}$

e)
ОДЗ: х >0; y > 0

{log₂ x + log₄ y = 4
{log₄ x + log₂ y = 5
  ║
  ∨
{log₂ x + 1/2 log₂ y = 4
{1/2log₂ x + log₂ y = 5
  ║
  ∨
{2log₂ x + log₂ y = 4 · 2
{log₂ x + 2log₂ y = 5 · 2
  ║
  ∨
{log₂ x² + log₂ y = 8
{log₂ x + log₂ y² = 10
  ║
  ∨
{log₂ x²y = 8
{log₂ xy² = 10
║
  ∨
{x²y = 2⁸
{xy² = 2¹⁰
Делим второе уравнение на первое:
xy²/x²y = 2¹⁰/2⁸
у/х= 2²

у/х = 4
у = 4х
Подставим у = 4х в уравнение х²у = 2⁸ и найдем х:
х² · 4х = 2⁸
4х³ = 2⁸
х³ = 2⁸ : 4
х³ = 2⁸ : 2²
х³ = 2⁶
х³ = 64
х = ∛64
х = 4

Найдём у = 4х   при х = 4
у = 4·4 = 16

Ответ: х=4; у = 16