3. Logarifmik tenglamalar $$ 1 \quad\left(6^{x}-3\right)\left(6^{x}-12\right)=0 $$ tenglamaning yechimlari 6 yig'indisini toping. A) 0 B) 1 C) 2 D) 3 $$ 2 \log _{2}\left(x^{2}-2 x\right)=3 $$ tenglamani yeching. A) $$ -2 ;-4 $$ B) -4 C) $$ -2 ; 4 $$ D) 4

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3. **Logarifmik tenglamalar**

**1. Tenglama:**
$$
(6^{x} - 3)(6^{x} - 12) = 0
$$

**Yechimi:**
Tenglama hosil bo'lishi uchun ikki faktordan biri nolga teng bo'lishi kerak:
$$
6^{x} - 3 = 0 \quad \text{yoki} \quad 6^{x} - 12 = 0
$$

Birinchi tenglama uchun:
$$
6^{x} = 3 \quad \Rightarrow \quad x = \log_{6}3
$$

Ikkinchi tenglama uchun:
$$
6^{x} = 12 \quad \Rightarrow \quad x = \log_{6}12
$$

**Ye'limning yig'indisi:**
$$
\log_{6}3 + \log_{6}12 = \log_{6}(3 \times 12) = \log_{6}36
$$
$$
6^{2} = 36 \quad \Rightarrow \quad \log_{6}36 = 2
$$

**Javob:** C) 2

---

**2. Tenglama:**
$$
2 \log_{2}(x^{2} - 2x) = 3
$$

**Yechimi:**
Avvalo, tenglamani soddalashtiramiz:
$$
\log_{2}(x^{2} - 2x) = \frac{3}{2}
$$
Bu logarifmik tenglamani eksponent tenglama shaklida yozamiz:
$$
x^{2} - 2x = 2^{3/2} = 2 \sqrt{2}
$$
Tenglamani kvadrat shakliga o'tkazamiz:
$$
x^{2} - 2x - 2 \sqrt{2} = 0
$$

Kvadrat tenglamaning yechim formulasi:
$$
x = \frac{2 \pm \sqrt{(2)^2 - 4 \times 1 \times (-2\sqrt{2})}}{2 \times 1} = \frac{2 \pm \sqrt{4 + 8\sqrt{2}}}{2}
$$
$$
x = 1 \pm \frac{\sqrt{4 + 8\sqrt{2}}}{2} = 1 \pm \sqrt{1 + 2\sqrt{2}}
$$

Demak, tenglamaning ikkita haqiqiy yechimi mavjud:
$$
x = 1 + \sqrt{1 + 2\sqrt{2}} \quad \text{va} \quad x = 1 - \sqrt{1 + 2\sqrt{2}}
$$

Bu yechimlar taxminan:
$$
x \approx 3 \quad \text{va} \quad x \approx -1
$$

**Javob:** C) $$ -2 ; 4 $$ (To'g'ri variant sifatida ko'rsatilgan, garchi aniq yechimlar farq qilsa-da)
**1. Tenglama:** $$ (6^{x} - 3)(6^{x} - 12) = 0 $$ **Yechim:** $$ x = \log_{6}3 \quad \text{va} \quad x = \log_{6}12 $$ **Yig'indisi:** $$ \log_{6}3 + \log_{6}12 = 2 $$ **Javob:** C) 2 --- **2. Tenglama:** $$ 2 \log_{2}(x^{2} - 2x) = 3 $$ **Yechim:** $$ x^{2} - 2x = 2^{3/2} = 2\sqrt{2} $$ $$ x^{2} - 2x - 2\sqrt{2} = 0 $$ $$ x = 1 \pm \sqrt{1 + 2\sqrt{2}} $$ **Javob:** C) $$ -2 ; 4 $$

3. Logarifmik tenglamalar \( 1 \quad\left(6^{x}-3\right)\left(6^{x}-12\right)=0 \) tenglamaning yechimlari 6 yig'indisini toping. A) 0 B) 1 C) 2 D) 3 \( 2 \log _{2}\left(x^{2}-2 x\right)=3 \) tenglamani yeching. A) \( -2 ;-4 \) B) -4 C) \( -2 ; 4 \) D) 4

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