Log_x²+x(x²-4)=log_4x²-6(x²-4). Created by kubsonone7. matematika-ru

Log_x²+x(x²-4)=log_4x²-6(x²-4)...

August 2022 1 1 Report

Log_x²+x(x²-4)=log_4x²-6(x²-4)

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Единорожек34 $log_{x^2+x}(x^2-4)=log_{4x^2-6}(x^2-4)$
ОДЗ:
x² + x > 0 ⇔ x(x + 1) > 0 ⇔ x∈(-∞;-1)∪(0;∞)
x² + x ≠ 1 ⇔ x² + x - 1 ≠ 0 ⇔ x ≠ (-1 ± √5)/2
x² - 4 > 0 ⇔ (x - 2)(x + 2) > 0 ⇔ x∈(-∞;-2)∪(2;∞)
4x² - 6 > 0 ⇔ x² - 6/4 > 0 ⇔ (x - √6 / 2)(x + √6 / 2) > 0 ⇔ x∈(-∞;-√6/2)∪(√6/2;∞)
4x² - 6 ≠ 1 ⇔ x ≠ ± √7 /2
Пересечение x∈(-∞;-2)∪(2;∞)

$log_{x^2+x}(x^2-4)-log_{4x^2-6}(x^2-4)=0\\\frac{lg(x^2-4)}{lg(x^2+x)}-\frac{lg(x^2-4)}{lg(4x^2-6)}=0\\\frac{lg(x^2-4)lg(4x^2-6)-lg(x^2-4)lg(x^2+x)}{lg(x^2+x)lg(4x^2-6)}=0\\lg(x^2-4) *(lg(4x^2-6) - lg(x^2+x))=0\\1) lg(x^2-4)=0\\x^2-4=1\\x=\pm \sqrt{5} \\2) lg(4x^2-6)-lg(x^2+x)=0\\ lg(4x^2-6)=lg(x^2+x)\\4x^2-6=x^2+x\\3x^2-x-6=0\\D=1+72=73\\x=\frac{1\pm \sqrt{73}}{6} \notin ODZ$
Ответ: $\pm \sqrt{5}$

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