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Kuis: super susahDiketahui pertidaksamaan8(logₐx)² + logₐ(x²) > 3(i) Tentukan hp

Berikut ini adalah pertanyaan dari xcvi pada mata pelajaran Matematika untuk jenjang Sekolah Menengah Atas

Kuis: super susahDiketahui pertidaksamaan
8(logₐx)² + logₐ(x²) > 3

(i) Tentukan hp untuk x
Jawab: hp = {x | ......., xER}

(ii) Tentukan hp untuk a
Jawab: hp = {a | ...... v ......, aER}

Jawaban dan Penjelasan

Berikut ini adalah pilihan jawaban terbaik dari pertanyaan diatas.

Jawab:

Penjelasan dengan langkah-langkah:

Catatan PENTING untuk para viewer : liat jawabannya di Brainly WEB jangan di APP

8\cdot \log^2_a(x)+2\log_a(x) > 3\\8\left(\log^2_a(x) + \dfrac{\log_a(x)}{4}\right) > 3\\8\left(\left(\log_a(x) + \dfrac{1}{8}\right)^2 - \dfrac{1}{64}\right) > 3\\8\left(\log_a(x) + \dfrac{1}{8}\right)^2 - \dfrac{1}{8} > 3\\64\left(\log_a(x) + \dfrac{1}{8}\right)^2 - 25 > 0\\\left(8\left(\log_a(x)+\dfrac{1}{8}\right) + 5\right)\left(8\left(\log_a(x)+\dfrac{1}{8}\right) - 5 \right) > 0\\\left(8\log_a(x)+ 6\right)\left(8\log_a(x)- 4 \right) > 0\\

\log_a(x^8\cdot a^6)\cdot \log_a\left(\dfrac{x^8}{a^4}\right) > 0\\\log_a(x^4\cdot a^3)\cdot \log_a\left(\dfrac{x^2}{a}\right) > 0\\ \to \left(\log_a(x^4\cdot a^3) \cap \log_a\left(\dfrac{x^2}{a}\right) > 0\right) \cup \left(\log_a(x^4\cdot a^3) \cap \log_a\left(\dfrac{x^2}{a}\right) < 0\right)\\

\text{Kasus 1 :}\\\log_a(x^4\cdot a^3) \cap \log_a\left(\dfrac{x^2}{a}\right) > 0 \to \log_a(x^4\cdot a^3) > 0\cap \log_a\left(\dfrac{x^2}{a}\right) > 0\\x^4\cdot a^3 > 1 \cap \dfrac{x^2}{a} > 1 \to \text{Syarat nilai : } x > 0, (0 < a < 1 )\cup (a > 1)\\x^4 > \dfrac{1}{a^3} \cap x^2 > a\\

\text{batas nilai $x$ : }\\\\\left(\left(x < -\dfrac{1}{\sqrt[4]{a^3}} \cup x > \dfrac{1}{\sqrt[4]{a^3}}\right) \cap \left( x < -\sqrt{a} \cap x > \sqrt{a} \right)\right)\cap \left( x > 0 \right)\cap \left( a > 0 \cap a\neq 1\right) \\\\\boxed{\boxed{ \left(x > \dfrac{1}{\sqrt[4]{a^3}}\cap x > \sqrt{a}\right) \cap ( a > 0\cap a\neq 1) }}

\text{Kasus 2 :}\\\log_a(x^4\cdot a^3) \cap \log_a\left(\dfrac{x^2}{a}\right) < 0 \to \log_a(x^4\cdot a^3) < 0\cap \log_a\left(\dfrac{x^2}{a}\right) < 0\\ (0 < x^4\cdot a^3 < 1) \cap \left( 0 < \dfrac{x^2}{a} < 1\right) \to \text{Syarat nilai : } x > 0, (0 < a < 1 )\cup (a > 1)\\\\ \left(\left(0 < x^4 < \dfrac{1}{a^3}\right) \cap \left( 0 < x^2 < a\right)\right) \cap \left(x > 0\right) \cap (a > 0 \cap a \neq 1)

\boxed{\boxed{ \left(x < \dfrac{1}{\sqrt[4]{a^3}} \cap x < \sqrt{a}\right) \cap ( a > 0\cap a\neq 1) }}

a)

Batas x :

\boxed{\boxed{ \left(\left(x > \dfrac{1}{\sqrt[4]{a^3}}\cap x > \sqrt{a}\right) \cup \left(x < \dfrac{1}{\sqrt[4]{a^3}} \cap x < \sqrt{a}\right) \right) \cap (x > 0)}}

Untuk 0 < a < 1 :

\boxed{\boxed{ \left( x < \sqrt{a}\cap (x < 1) \right) \cup \left(x > \dfrac{1}{\sqrt[4]{a^3}}\cap \left(x > 1\right)\right) }}

a > 1 :

\boxed{\boxed{ \left(x < \dfrac{1}{\sqrt[4]{a^3}} \cap (0 < x < 1)\right)\cup \left(x > \sqrt{a} \cap \left(x > 1\right)\right) }}

b)

Batas a :

\left(x^4 > \dfrac{1}{a^3} \cap x^2 > a \right) \cup \left((0 < x^4\cdot a^3 < 1) \cap \left( 0 < \dfrac{x^2}{a} < 1\right) \right)\\\\

\boxed{\boxed{\left( \left(a > \dfrac{1}{x\sqrt[3]{x}} \cap a < x^2\right)\cup \left( a < \dfrac{1}{x\sqrt[3]{x}}\cap a > x^2 \right)\right) \cap \left( a > 0 \cap a\neq 1 \right)}}

Untuk 0 < x < 1 :

\boxed{\boxed{ (a > x^2\cap \left(a < 1\right)) \cup \left(a < \dfrac{1}{x\sqrt[3]{x}} \cap \left(a > 1\right)\right) }}

x > 1 :

\boxed{\boxed{ \left(a > \dfrac{1}{x\sqrt[3]{x}} \cap \left( 0 < a < 1\right)\right)\cup (a < x^2\cap \left(a > 1\right)) \right) }}

Jawab:Penjelasan dengan langkah-langkah:Catatan PENTING untuk para viewer : liat jawabannya di Brainly WEB jangan di APP[tex]8\cdot \log^2_a(x)+2\log_a(x) > 3\\8\left(\log^2_a(x) + \dfrac{\log_a(x)}{4}\right) > 3\\8\left(\left(\log_a(x) + \dfrac{1}{8}\right)^2 - \dfrac{1}{64}\right) > 3\\8\left(\log_a(x) + \dfrac{1}{8}\right)^2 - \dfrac{1}{8} > 3\\64\left(\log_a(x) + \dfrac{1}{8}\right)^2 - 25 > 0\\\left(8\left(\log_a(x)+\dfrac{1}{8}\right) + 5\right)\left(8\left(\log_a(x)+\dfrac{1}{8}\right) - 5 \right) > 0\\\left(8\log_a(x)+ 6\right)\left(8\log_a(x)- 4 \right) > 0\\[/tex][tex]\log_a(x^8\cdot a^6)\cdot \log_a\left(\dfrac{x^8}{a^4}\right) > 0\\\log_a(x^4\cdot a^3)\cdot \log_a\left(\dfrac{x^2}{a}\right) > 0\\ \to \left(\log_a(x^4\cdot a^3) \cap \log_a\left(\dfrac{x^2}{a}\right) > 0\right) \cup \left(\log_a(x^4\cdot a^3) \cap \log_a\left(\dfrac{x^2}{a}\right) < 0\right)\\[/tex][tex]\text{Kasus 1 :}\\\log_a(x^4\cdot a^3) \cap \log_a\left(\dfrac{x^2}{a}\right) > 0 \to \log_a(x^4\cdot a^3) > 0\cap \log_a\left(\dfrac{x^2}{a}\right) > 0\\x^4\cdot a^3 > 1 \cap \dfrac{x^2}{a} > 1 \to \text{Syarat nilai : } x > 0, (0 < a < 1 )\cup (a > 1)\\x^4 > \dfrac{1}{a^3} \cap x^2 > a\\[/tex][tex]\text{batas nilai $x$ : }\\\\\left(\left(x < -\dfrac{1}{\sqrt[4]{a^3}} \cup x > \dfrac{1}{\sqrt[4]{a^3}}\right) \cap \left( x < -\sqrt{a} \cap x > \sqrt{a} \right)\right)\cap \left( x > 0 \right)\cap \left( a > 0 \cap a\neq 1\right) \\\\\boxed{\boxed{ \left(x > \dfrac{1}{\sqrt[4]{a^3}}\cap x > \sqrt{a}\right) \cap ( a > 0\cap a\neq 1) }}[/tex][tex]\text{Kasus 2 :}\\\log_a(x^4\cdot a^3) \cap \log_a\left(\dfrac{x^2}{a}\right) < 0 \to \log_a(x^4\cdot a^3) < 0\cap \log_a\left(\dfrac{x^2}{a}\right) < 0\\ (0 < x^4\cdot a^3 < 1) \cap \left( 0 < \dfrac{x^2}{a} < 1\right) \to \text{Syarat nilai : } x > 0, (0 < a < 1 )\cup (a > 1)\\\\ \left(\left(0 < x^4 < \dfrac{1}{a^3}\right) \cap \left( 0 < x^2 < a\right)\right) \cap \left(x > 0\right) \cap (a > 0 \cap a \neq 1)[/tex][tex]\boxed{\boxed{ \left(x < \dfrac{1}{\sqrt[4]{a^3}} \cap x < \sqrt{a}\right) \cap ( a > 0\cap a\neq 1) }}[/tex]a) Batas x :[tex]\boxed{\boxed{ \left(\left(x > \dfrac{1}{\sqrt[4]{a^3}}\cap x > \sqrt{a}\right) \cup \left(x < \dfrac{1}{\sqrt[4]{a^3}} \cap x < \sqrt{a}\right) \right) \cap (x > 0)}}[/tex]Untuk 0 < a < 1 :[tex]\boxed{\boxed{ \left( x < \sqrt{a}\cap (x < 1) \right) \cup \left(x > \dfrac{1}{\sqrt[4]{a^3}}\cap \left(x > 1\right)\right) }}[/tex]a > 1 :[tex]\boxed{\boxed{ \left(x < \dfrac{1}{\sqrt[4]{a^3}} \cap (0 < x < 1)\right)\cup \left(x > \sqrt{a} \cap \left(x > 1\right)\right) }}[/tex]b) Batas a :[tex]\left(x^4 > \dfrac{1}{a^3} \cap x^2 > a \right) \cup \left((0 < x^4\cdot a^3 < 1) \cap \left( 0 < \dfrac{x^2}{a} < 1\right) \right)\\\\[/tex][tex]\boxed{\boxed{\left( \left(a > \dfrac{1}{x\sqrt[3]{x}} \cap a < x^2\right)\cup \left( a < \dfrac{1}{x\sqrt[3]{x}}\cap a > x^2 \right)\right) \cap \left( a > 0 \cap a\neq 1 \right)}}[/tex]Untuk 0 < x < 1 :[tex]\boxed{\boxed{ (a > x^2\cap \left(a < 1\right)) \cup \left(a < \dfrac{1}{x\sqrt[3]{x}} \cap \left(a > 1\right)\right) }}[/tex]x > 1 :[tex]\boxed{\boxed{ \left(a > \dfrac{1}{x\sqrt[3]{x}} \cap \left( 0 < a < 1\right)\right)\cup (a < x^2\cap \left(a > 1\right)) \right) }}[/tex]

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Last Update: Fri, 30 Sep 22
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