3. LOGARIFMIK TENGLAMALAR 11. \( \log _{x}(4 x-3)=2 \) tenglama ildizlari yig'indisini toping. A) 2 B) 3 C) 4 D) 5 \( 12 \log _{7}(2 x-5)=\log _{7}(x-3) \) tenglamani yeching. A) 0 B) 2 C) 3 D) 4 \( 13(x-2) \log _{3}\left(x^{2}-24\right)=0 \) tenglamaning haqiqiy ildizlari nechta? A) 0 B) 1 C) 2 D) 3 \( 14 \log _{2}(x-8)+\log _{3} 27=4 \) tenglama ildizi \( x_{0} \) bo'lsa, \( x_{0}+5 \) ning qiymatini toping. A) 10 B) 12 C) 15 D) 17

Quyidagi logarifmik tenglamalarning yechimlarini ko'rib chiqamiz va to'g'ri javoblarni aniqlaymiz. --- ### **11. \( \log _{x}(4 x-3)=2 \) tenglama ildizlari yig'indisini toping.** **Yechim:** \[ \log_{x}(4x - 3) = 2 \implies x^2 = 4x - 3 \implies x^2 - 4x + 3 = 0 \] Kvadrat tenglama ildizlari: \[ x = \frac{4 \pm \sqrt{16 - 12}}{2} = \frac{4 \pm 2}{2} \implies x_1 = 3, \quad x_2 = 1 \] **Tekshiruv:** - \( x = 3 \): Logarifm asos \(3 > 0\) va \( \neq 1 \), argument \(4(3) - 3 = 9 > 0\). Qabul qilinadi. - \( x = 1 \): Logarifm asos \(1\) bo'lishi mumkin emas. Rad etiladi. **Natija:** Faqat \( x = 3 \) haqiqiy yechim. **To'g'ri javob:** **B) 3** --- ### **12. \( 12 \log _{7}(2 x-5)=\log _{7}(x-3) \) tenglamani yeching.** **Yechim:** \[ 12 \log_{7}(2x - 5) = \log_{7}(x - 3) \] \[ (2x - 5)^{12} = x - 3 \] Bu tenglama uchun \( x > \frac{5}{2} \) va \( x > 3 \). Grafiklarni tahlil qilganda, chap tomonda \( (2x - 5)^{12} \) juda tez o'sadi va har doim \( x - 3 \) dan katta bo'ladi. Demak, haqiqatda hech qanday yechim mavjud emas. **Natija:** Hech qanday yechim yo'q. **To'g'ri javob:** **A) 0** --- ### **13. \( (x-2) \log _{3}\left(x^{2}-24\right)=0 \) tenglamaning haqiqiy ildizlari nechta?** **Yechim:** \[ (x - 2) \log_{3}(x^2 - 24) = 0 \] Bu tenglama quyidagi hollarda to'g'ri: 1. \( x - 2 = 0 \implies x = 2 \) (tekshirish kerak) 2. \( \log_{3}(x^2 - 24) = 0 \implies x^2 - 24 = 1 \implies x^2 = 25 \implies x = \pm 5 \) **Tekshiruv:** - \( x = 2 \): \( x^2 - 24 = 4 - 24 = -20 < 0 \). Qabul qilinmaydi. - \( x = 5 \): \( 5^2 - 24 = 1 > 0 \). Qabul qilinadi. - \( x = -5 \): \( (-5)^2 - 24 = 1 > 0 \). Qabul qilinadi. **Natija:** Ikkala yechim \( x = 5 \) va \( x = -5 \) mavjud. **To'g'ri javob:** **C) 2** --- ### **14. \( \log _{2}(x-8)+\log _{3} 27=4 \) tenglama ildizi \( x_{0} \) bo'lsa, \( x_{0}+5 \) ning qiymatini toping.** **Yechim:** \[ \log_{2}(x - 8) + \log_{3}(27) = 4 \] \[ \log_{3}(27) = 3 \implies \log_{2}(x - 8) + 3 = 4 \implies \log_{2}(x - 8) = 1 \] \[ x - 8 = 2^1 = 2 \implies x = 10 \] Shundan \( x_0 = 10 \), demak \( x_0 + 5 = 15 \). **To'g'ri javob:** **C) 15** --- **Javoblar:** 11. **B) 3** 12. **A) 0** 13. **C) 2** 14. **C) 15**