766. 1) log2 (sqrt(2)); 2) log3 (1/sqrt(3)); 3) log0.5 (1/sqrt(32)); 4) log7 (sqrt(7)/49); 5) log128(64); 6) log27(243); 7) log64(256); 8) log81(27); 9) log(sqrt(1/(3*cbrt(3)))); 10) log1/2 (1/(4*sqrt(2)))

766. Вычисление логарифмов

1. 1) log₂ √2
   Запишем √2 как 2^1/2. Тогда log₂ 2^1/2 = 1/2.
2. 2) log₃ (1/∛3)
   Запишем ∛3 как 3^1/3, тогда 1/∛3 = 3^-1/3. log₃ 3^-1/3 = -1/3.
3. 3) log_0,5 (1/√32)
   0,5 = 1/2 = 2^-1. 32 = 2⁵, √32 = 2^5/2, 1/√32 = 2^-5/2.
   log_2^-1 2^-5/2 = –5/2 / –1 = 5/2.
4. 4) log₇ (√7 / 49)
   √7 = 7^1/2. 49 = 7².
   √7 / 49 = 7^1/2 / 7² = 7^{1/2 - 2} = 7^-3/2.
   log₇ 7^-3/2 = -3/2.
5. 5) log₁₂₈ 64
   128 = 2⁷, 64 = 2⁶.
   log_2⁷ 2⁶ = 6/7.
6. 6) log₂₇ 243
   27 = 3³, 243 = 3⁵.
   log_3³ 3⁵ = 5/3.
7. 7) log₆₄ 256
   64 = 4³, 256 = 4⁴.
   log_4³ 4⁴ = 4/3.
8. 8) log₈₁ 27
   81 = 3⁴, 27 = 3³.
   log_3⁴ 3³ = 3/4.
9. 9) log_√ (1/(3∛3))
   √ = 3^1/2. 3∛3 = 3 * 3^1/3 = 3^4/3.
   1/(3∛3) = 3^-4/3.
   log_3^1/2 3^-4/3 = –4/3 / (1/2) = -8/3.
10. 10) log_1/2 (1/(4√2))
    1/2 = 2^-1. 4√2 = 2² * 2^1/2 = 2^5/2.
    1/(4√2) = 2^-5/2.
    log_2^-1 2^-5/2 = –5/2 / –1 = 5/2.