7, *,*,*,*,*,*,35
а1, ..........,а8
а8=а1+7d
7+7d=35
7d=28
d=4
7, 11, 15, 19, 23, 27, 31, 35
С первой системы: х=3+у
подставляешь во вторую систему: (3+у)у - 4=0
получается у в квадрате +3у - 4=0
у1 = -4, у2 = 1
х1=-1, х2=4.
Ответ: (-1;-4), (4;1).
2.
16^(log₄ 3)= 4^(2log₄ 3) = 4^(log₄ 3²)= 3² =9
3. log₀.₅ 64 = log₁/₂ 2⁶ = log₁/₂ (1/2)⁻⁶ = -6
4. log₂ 16=log₂ 2⁴ =4
5. log₈ 512 * log₂ 32 = log₈ 8³ * log₂ 2⁵ = 3*5=15
6. log₁₂ 252 - log₁₂ 1.75 = log₁₂ (252/1.75)= log₁₂ 144 = log₁₂ 12² =2
7. log₅ 5 + log₀.₂₅ 64 = 1 + log₁/₄ 4³ = 1+log₁/₄ (1/4)⁻³ =1-3= -2
8. log₁.₈ 5 - log₁.₈ 9 = log₁.₈ (5/9)=log₉/₅ (9/5)⁻¹ = -1
9. log₀.₈ 3 * log₃ 1.25 = <u> 1 </u> * log₃ 1.25 =<u> log₃ 1.25 </u>= log₀.₈ 1.25 =
log₃ 0.8 log₃ 0.8
= log₈/₁₀ (10/8)=log₈/₁₀ (8/10)⁻¹ = -1
10. <u>log₉ 10 </u>+ log₁₁ 0.1 = log₁₁ 10 + log₁₁ 0.1 =log₁₁ (10*0.1)=log₁₁ 1 =0
log₉ 11
11. log₂ 5 * log₅ 8 = <u>log₅ 8 </u>= log₂ 8 = log₂ 2³ = 3
log₅ 2
12. log₂ 3.2 + log₂ 5=log₂ (3.2*5)=log₂ 16 = log₂ 2⁴= 4
13. (1-log₆ 54)(1-log₉ 54)=(log₆ 6 - log₆ 54)(log₉ 9 - log₉ 54)=
= log₆ (6/54) * log₉ (9/54) = log₆ (1/9) * log₉ (1/6) =
=<u> log₉ (1/6) </u>= -<u> log₁/₉ (1/6) </u>= <u>log₁/₉ (1/6)⁻¹ </u>= <u>log₁/₉ 6 </u>= 1
log₁/₉ 6 log₁/₉ 6 log₁/₉ 6 log₁/₉ 6
14. 100 log₆ ⁴√6 = 100 * (¹/₄) log₆ 6 = 25*1=25
15. log(⁵√12) 12 = 1/¹/₅ log₁₂ 12 = 5*1 =5
16. <u> log₂ 48 </u> = <u> log₂ 48 </u> = <u> log₂ 48 </u> = <u>log₂ 48 </u>= 1
3+log₂ 6 log₂ 2³ + log₂ 6 log₂ (8*6) log₂ 48
17. <u>log₃ 14 </u>=<u> log₃ 14 </u>= <u> log₃ 14 </u> = <u> 1 </u>= 2
log₉ 14 log₃² 14 ¹/₂ log₃ 14 ¹/₂
18. <u>log₆ 81 </u>= log₉ 81 = log₉ 9² = 2
log₆ 9
19. 5^(log₂₅ 49) =5^(log₅² 7²) = 5^(²/₂ log₅ 7)= 5^(log₅ 7) = 7
20. log²√₈ 64 = log² (8^(¹/₂)) (64²)^(¹/₂) = log²₈ 64² =
= log²₈ 8⁴ = 4² = 16
21. 9^(3log₉ 11)=9^(log₉ 11³) = 11³ = 1331
22. 4^(log₂ √3) = 2^(2log₂ √3)= 2^(log₂ (√3)²) = 2^(log₂ 3) = 3
23. log₄ log₉ 81 = log₄ log₉ 9² = log₄ 2 = log₂² 2 = ¹/₂ log₂ 2 = 1/2
24. <u> 60 </u> =<u> 60 </u> = 6
4^(log₄ 10) 10
25. log₁/₈ √8 = log₈⁻¹ 8^(¹/₂) = -¹/₂ log₈ 8 = -1/2
26. <u>4^(log₆ 72)</u> = 4^(log₆ 72 - log₆ 2) = 4^(log₆ (72/2)) = 4^(log₆ 36) =
4^(log₆ 2)
= 4² = 16
27.<u> log₃ √5 </u>= log₅ √5 = 1/2
log₃ 5
28. (5^(log₃ 7))^log₇ 3 = 5^(log₃ 7 * log₇ 3) = 5^(log₇ 3/log₇ 3)=5¹ =5
Cos2a+cos14a+cos6a+cos10a=2(cos a+cos7a+cos3a+cos5a)
sin4b+sin10b+sin22b+sin16b=2(sin2b+sin5b+sin11b+sin8b)
