1. Muammo: Berilgan ifodalardan eng kattasini toping: A) $\log_2 18 - \log_2 9$ B) $3^{\log_3 6}$ C) $\lg 25 + \lg 4$ D) $\log_{13} 169^2$ E) $\log_h 64^3$ 2. Formulalar va qoidalar: - Logarifmning farqi: $\log_a b - \log_a c = \log_a \frac{b}{c}$ - Eksponent va logarifmning o'zaro bog'lanishi: $a^{\log_a b} = b$ - Logarifmning yig'indisi: $\lg a + \lg b = \lg (ab)$ - Logarifmning kuchi: $\log_a b^n = n \log_a b$ 3. Hisoblashlar: A) $\log_2 18 - \log_2 9 = \log_2 \frac{18}{9} = \log_2 2 = 1$ B) $3^{\log_3 6} = 6$ C) $\lg 25 + \lg 4 = \lg (25 \times 4) = \lg 100 = 2$ D) $\log_{13} 169^2 = \log_{13} (13^2)^2 = \log_{13} 13^4 = 4$ E) $\log_h 64^3 = 3 \log_h 64$ (qiymati noma'lum, chunki $h$ berilmagan, shuning uchun uni hisoblay olmaymiz) 4. Natijalar: A) 1 B) 6 C) 2 D) 4 E) noma'lum Eng katta qiymat: B) 6 --- 21. Hisoblang: $343^{\log_{49} 4}$ 1) $343 = 7^3$, $49 = 7^2$ 2) $343^{\log_{49} 4} = (7^3)^{\log_{7^2} 4} = 7^{3 \cdot \log_{7^2} 4}$ 3) $\log_{7^2} 4 = \frac{\log_7 4}{\log_7 7^2} = \frac{\log_7 4}{2}$ 4) Shunday qilib, $$7^{3 \cdot \frac{\log_7 4}{2}} = 7^{\frac{3}{2} \log_7 4} = (7^{\log_7 4})^{\frac{3}{2}} = 4^{\frac{3}{2}} = (\sqrt{4})^3 = 2^3 = 8$$ Javob: A) 8 --- 22. Sonlarni kamayish tartibida joylashtiring: $n = \log_{1/4} + \log_{1/2}$ (bu ifoda to'liq emas, ehtimol $\log_{1/4} x + \log_{1/2} y$ bo'lishi kerak, lekin $x,y$ berilmagan, shuning uchun taxmin qilamiz) $m = \log_2 15 - \log_{1/5}$ (bu ham to'liq emas, ehtimol $\log_2 15 - \log_{1/5} z$) $p = \ln e^{-2} = -2$ Agar $n$ va $m$ qiymatlari aniq bo'lmasa, faqat $p = -2$ ni bilamiz. Shuning uchun bu savolga aniq javob berish uchun qo'shimcha ma'lumot kerak. --- 23. Hisoblang: $\log_2 \sqrt{512}$ 1) $\sqrt{512} = 512^{1/2}$ 2) $512 = 2^9$ 3) $\log_2 512^{1/2} = \frac{1}{2} \log_2 2^9 = \frac{1}{2} \times 9 = 4.5$ Javob variantlarda yo'q, lekin eng yaqin qiymat 4 yoki 6. To'g'ri javob $4.5$. --- 24. Hisoblang: $\log_3 4 \cdot \log_5 \cdot \log_6 \cdot \log_7 \cdot \log_7^8 \cdot \log_9$ Bu ifoda to'liq emas, logarifm bazalari va argumentlari ko'rsatilmagan, shuning uchun hisoblash mumkin emas. --- 25. Hisoblang: $\log_3 2 \cdot \log_4 3 \cdot \log_4 \cdot \log_5 \cdot \log_6 \cdot \log_7 \cdot \log_{s}7$ Bu ifoda ham to'liq emas, logarifm bazalari va argumentlari ko'rsatilmagan. --- 26. Hisoblang: $2^{\log_2 3} \cdot \log_3 2 \cdot \log_3 \frac{1}{81}$ 1) $2^{\log_2 3} = 3$ 2) $\log_3 2$ qoldi 3) $\log_3 \frac{1}{81} = \log_3 81^{-1} = -\log_3 81 = -4$ (chunki $81 = 3^4$) 4) Shunday qilib, $$3 \cdot \log_3 2 \cdot (-4) = -12 \log_3 2$$ $\log_3 2$ ni taxminan 0.63 deb olsak, $$-12 \times 0.63 = -7.56$$ Eng yaqin javob: D) -8 --- 27. Hisoblang: $\sqrt{25}^{\log_{36} 5} + 49^{\log_5 7}$ 1) $\sqrt{25} = 5$ 2) $5^{\log_{36} 5} = 5^{\frac{\log 5}{\log 36}}$ 3) $49^{\log_5 7} = (7^2)^{\log_5 7} = 7^{2 \log_5 7} = (7^{\log_5 7})^2 = 5^2 = 25$ 4) $5^{\log_{36} 5} = 36^{\log_{36} 5} = 5$ 5) Shunday qilib, $$5 + 25 = 30$$ Javob variantlarda yo'q, lekin eng yaqin 14 emas. --- 28. Soddalashtiring: $36^{\log_6 5} + 10^{-1 \cdot \lg 2} - 3^{\log_6 36}$ 1) $36 = 6^2$ 2) $36^{\log_6 5} = (6^2)^{\log_6 5} = 6^{2 \log_6 5} = (6^{\log_6 5})^2 = 5^2 = 25$ 3) $10^{-1 \cdot \lg 2} = 10^{-\lg 2} = \frac{1}{10^{\lg 2}} = \frac{1}{2}$ 4) $3^{\log_6 36} = 3^{\log_6 6^2} = 3^{2 \log_6 6} = 3^2 = 9$ 5) Jamlaymiz: $$25 + \frac{1}{2} - 9 = 16.5$$ Javob variantlarda yo'q, lekin eng yaqin 21 emas. --- 29. Soddalashtiring: $$\frac{\log_2 14 + \log_4 14 \log_7 - 2 \log_2 7}{\log_2 14 + 2 \log_7 7}$$ Bu ifoda to'liq emas, $\log_4 14 \log_7$ va $\log_7 7$ ning argumentlari ko'rsatilmagan. --- 30. Hisoblang: $$(\log_3 27 - \log_3 9) \cdot (\log_3 48 + \log_3 \frac{1}{16}) + \log_3 81$$ 1) $\log_3 27 = 3$ 2) $\log_3 9 = 2$ 3) $\log_3 48 + \log_3 \frac{1}{16} = \log_3 \left(48 \times \frac{1}{16}\right) = \log_3 3 = 1$ 4) $\log_3 81 = 4$ 5) Hisoblaymiz: $$(3 - 2) \times 1 + 4 = 1 + 4 = 5$$ Javob variantlarda yo'q. --- "slug":"logarithm problems","subject":"algebra","desmos":{"latex":"","features":{"intercepts":true,"extrema":true}},"q_count":10