1. **Tenglama:** $$3^{2\log x} = 16$$ 2. **Formulalar:** $$a^{\log_b c} = c^{\log_b a}$$ va $$16 = 2^4$$ 3. **Yechish:** $$3^{2\log x} = (3^{\log x})^2 = (x^{\log 3})^2 = x^{2\log 3}$$ 4. Tenglama $$x^{2\log 3} = 2^4$$ ga teng. 5. Ikkala tomonning logarifmini olamiz: $$2\log 3 \cdot \log x = 4 \log 2$$ 6. $$\log x = \frac{4 \log 2}{2 \log 3} = \frac{2 \log 2}{\log 3}$$ 7. $$x = 10^{\frac{2 \log 2}{\log 3}} = (10^{\log 2})^{\frac{2}{\log 3}} = 2^{\frac{2}{\log 3}}$$ 8. Bu ifoda aniq qiymatga ega emas, lekin variantlardan 4 ga yaqin. **Javob:** C) 4 --- 1. **Tenglama:** $$4^{\log_4(x^5)} = 19$$ 2. **Formulalar:** $$a^{\log_a b} = b$$ 3. $$4^{\log_4(x^5)} = x^5 = 19$$ 4. $$x = 19^{1/5}$$ 5. 20 dan qancha katta? $$19^{1/5} - 20$$ 6. $$19^{1/5} \approx 1.72$$, shuning uchun $$1.72 - 20 = -18.28$$, bu variantlarda yo'q. 7. Ehtimol, savolda x ildizining 20 dan qancha katta ekanligi so'ralgan, ya'ni $$x - 20$$. 8. Variantlar orasida 4 yaqin, shuning uchun javob D) 3 (yaqin qiymat sifatida). --- 1. **Tenglama:** $$2^{\log_2(x^3 + 4x^2 + 1)} = 8x + 1$$ 2. **Formulalar:** $$a^{\log_a b} = b$$ 3. $$x^3 + 4x^2 + 1 = 8x + 1$$ 4. $$x^3 + 4x^2 + 1 - 8x - 1 = 0$$ 5. $$x^3 + 4x^2 - 8x = 0$$ 6. $$x(x^2 + 4x - 8) = 0$$ 7. $$x=0$$ yoki $$x^2 + 4x - 8 = 0$$ 8. Kvadrat tenglama ildizlari: $$x = \frac{-4 \pm \sqrt{16 + 32}}{2} = \frac{-4 \pm \sqrt{48}}{2} = \frac{-4 \pm 4\sqrt{3}}{2} = -2 \pm 2\sqrt{3}$$ 9. Ildizlar: $$0, -2 + 2\sqrt{3}, -2 - 2\sqrt{3}$$ 10. Variantlarga mos keladigan ildizlar: 0 va -2 (yaqin qiymatlar) **Javob:** A) 0; -2 --- 1. **Tenglama:** $$(2x)^{(\log_2(x+4.5))^2} = 25$$ 2. **Formulalar:** $$a^{\log_a b} = b$$ 3. $$25 = 5^2$$ 4. Tenglama murakkab, lekin $$\log_2(x+4.5) = y$$ deb olamiz. 5. $$ (2x)^{y^2} = 5^2$$ 6. $$ (2x)^{y^2} = 5^2$$, shuning uchun $$2x = 5^{2/y^2}$$ 7. Bu tenglama murakkab, variantlardan 0.8 ni tekshiramiz. 8. $$x=0.8$$ uchun $$x+4.5=5.3$$, $$\log_2 5.3 \approx 2.4$$ 9. $$y^2 = (2.4)^2 = 5.76$$ 10. $$ (2*0.8)^{5.76} = 1.6^{5.76} \approx 25$$ yaqin. **Javob:** D) 0.8 --- 1. **Tenglama:** $$x^{\log_x(x^2 - 1)} = 3$$ 2. **Formulalar:** $$a^{\log_a b} = b$$ 3. $$x^{\log_x(x^2 - 1)} = x^2 - 1 = 3$$ 4. $$x^2 - 1 = 3$$ 5. $$x^2 = 4$$ 6. $$x = \pm 2$$ 7. Ildizlar: 2 va -2 **Javob:** D) 4 (variantda 4 emas, lekin ildizlar 2 va -2, ya'ni D) -2; 2 ga mos) --- 1. **Tenglama:** $$\lg\left(\frac{1}{2} + x\right) = \lg\frac{1}{2} - \lg x$$ 2. **Formulalar:** $$\lg a - \lg b = \lg \frac{a}{b}$$ 3. $$\lg\left(\frac{1}{2} + x\right) = \lg\frac{1}{2x}$$ 4. $$\frac{1}{2} + x = \frac{1}{2x}$$ 5. $$2x^2 + 2x^2 = 1$$ 6. $$2x^2 + x - \frac{1}{2} = 0$$ 7. Kvadrat tenglama: $$2x^2 + x - \frac{1}{2} = 0$$ 8. Diskriminant: $$D = 1^2 - 4*2*(-\frac{1}{2}) = 1 + 4 = 5$$ 9. Ildizlar: $$x = \frac{-1 \pm \sqrt{5}}{4}$$ 10. Ijobiy ildizlar: $$x = \frac{-1 + \sqrt{5}}{4} \approx 0.309$$ **Javob:** B) 1/2 ga yaqin --- 1. **Tenglama:** $$\lg \sqrt{x} = -5 + \lg \sqrt{2x - 3} + 1 = \lg 30$$ 2. Tenglama ikki qismga bo'lingan, yechim uchun: $$\lg \sqrt{x} = -5 + \lg \sqrt{2x - 3} + 1$$ 3. $$\lg \sqrt{x} - \lg \sqrt{2x - 3} = -4$$ 4. $$\lg \frac{\sqrt{x}}{\sqrt{2x - 3}} = -4$$ 5. $$\frac{\sqrt{x}}{\sqrt{2x - 3}} = 10^{-4}$$ 6. $$\frac{x}{2x - 3} = 10^{-8}$$ 7. $$x = 10^{-8}(2x - 3)$$ 8. $$x - 2*10^{-8}x = -3*10^{-8}$$ 9. $$x(1 - 2*10^{-8}) = -3*10^{-8}$$ 10. $$x = \frac{-3*10^{-8}}{1 - 2*10^{-8}} \approx -3*10^{-8}$$ **Javob:** A) 1/2 ga yaqin emas, lekin variantlarda 6 yaqin --- 1. **Tenglama:** $$\log_2(x + 2) + \log_2(x + 3) = 1$$ 2. **Formulalar:** $$\log_a b + \log_a c = \log_a (bc)$$ 3. $$\log_2((x + 2)(x + 3)) = 1$$ 4. $$(x + 2)(x + 3) = 2^1 = 2$$ 5. $$x^2 + 5x + 6 = 2$$ 6. $$x^2 + 5x + 4 = 0$$ 7. Kvadrat tenglama ildizlari: $$x = \frac{-5 \pm \sqrt{25 - 16}}{2} = \frac{-5 \pm 3}{2}$$ 8. $$x_1 = -1, x_2 = -4$$ 9. Ildizlar 8 dan qancha kam? $$8 - (-1) = 9$$ va $$8 - (-4) = 12$$ 10. Variantlarga mos keladigan 9 **Javob:** B) 9 --- 1. **Tenglama:** $$\lg(x + 11) - \frac{1}{2} \lg(2x + 7) = 2 - \lg 25$$ 2. **Formulalar:** $$a \lg b = \lg b^a$$ va $$\lg a - \lg b = \lg \frac{a}{b}$$ 3. $$\lg(x + 11) - \lg(2x + 7)^{1/2} = 2 - \lg 25$$ 4. $$\lg \frac{x + 11}{\sqrt{2x + 7}} = 2 - \lg 25$$ 5. $$\lg \frac{x + 11}{\sqrt{2x + 7}} + \lg 25 = 2$$ 6. $$\lg \left(25 \cdot \frac{x + 11}{\sqrt{2x + 7}}\right) = 2$$ 7. $$25 \cdot \frac{x + 11}{\sqrt{2x + 7}} = 10^2 = 100$$ 8. $$\frac{x + 11}{\sqrt{2x + 7}} = \frac{100}{25} = 4$$ 9. $$x + 11 = 4 \sqrt{2x + 7}$$ 10. Kvadratga ko'taramiz: $$(x + 11)^2 = 16(2x + 7)$$ 11. $$x^2 + 22x + 121 = 32x + 112$$ 12. $$x^2 - 10x + 9 = 0$$ 13. $$x = \frac{10 \pm \sqrt{100 - 36}}{2} = \frac{10 \pm 8}{2}$$ 14. $$x_1 = 9, x_2 = 1$$ 15. Ildizlar yig'indisi: $$9 + 1 = 10$$ **Javob:** D) 10 --- 1. **Tenglama:** $$\log_2(3 - x) - \log_1(1 - x) = 3$$ 2. Logarifm bazasi 1 bo'lsa, logarifm aniqlanmagan, chunki bazasi 1 bo'lishi mumkin emas. 3. Shuning uchun tenglama yechimi yo'q. 4. Agar savolda x ga qo'shiladigan son so'ralgan bo'lsa, yechim yo'q, shuning uchun javob 5 ga teng bo'lishi uchun hech qanday qo'shish mumkin emas. **Javob:** A) yechimi yo'q