767. 1) 9^(2*log3(5)); 2) (1/9)^(-log3(4)); 3) (1/4)^(-5*log2(3)); 4) 27^(-4*log3(5)); 5) 10^(3-log10(5)); 6) (1/7)^(1+2*log7(3))

767. Вычисление выражений со степенями и логарифмами

1. 1) 9^2log₃5
   9 = 3². Значит, 9^2log₃5 = (3²)^2log₃5 = 3^4log₃5 = 3^log₃54 = 5⁴ = 625.
2. 2) (1/9)^-log₃4
   1/9 = 3^-2. Тогда (3^-2)^-log₃4 = 3^2log₃4 = 3^log₃42 = 4² = 16.
3. 3) (1/4)^-5log₂3
   1/4 = 2^-2. Тогда (2^-2)^-5log₂3 = 2^10log₂3 = 2^log₂310 = 3¹⁰ = 59049.
4. 4) 27^-4log₃5
   27 = 3³. Тогда (3³)^-4log₃5 = 3^-12log₃5 = 3^log₃5-12 = 5^-12 = 1 / 5¹².
5. 5) 10^3-log₁₀5
   10^3-log₁₀5 = 10³ / 10^log₁₀5 = 1000 / 5 = 200.
6. 6) (1/7)^1+2log₇3
   (1/7)^1+2log₇3 = (1/7)¹ * (1/7)^2log₇3 = (1/7) * (7^-1)^2log₇3 = (1/7) * 7^-2log₇3 = (1/7) * 7^log₇3-2 = (1/7) * 3^-2 = (1/7) * (1/9) = 1/63.