Решить систему уравнений:log₈ (xy) = 3log₈x*log₈y; 4log₈ (x/y) = (log₈x) / (log₈y)

{log₃ (xy) = 3log₃x * log₃y,

{4log₃ (x/y) = (log₃x) / (log₃y).

{log₃x + log₃y = 3log₃x * log₃y,

{4log₃x - 4log₃y = (log₃x) / (log₃y).

Пусть log₃x = m, log₃y = n, тогда:

{m + n = 3m * n,

{4m - 4n = m/n.

m = n / (3n - 1),

4n / (3n - 1) - 4n = n / (3n - 1) : n,

4n - 12n² + 4n = 1,

- 12n² + 8n - 1 = 0,

D = 16,

n₁ = ( - 8 + 4) / ( - 2 * 12) = 1/6,

n₂ = 1/2,

m₁ = 1/6 : (3/6 - 1) = - 1/3.

m₂ = 1/2 : (3/2 - 1) = 1.

log₃x = - 1/3,

х₁ = 3 ^{- 1/3} = 1/³√3.

log₃y = 1/6,

у₁ = ⁶√3.

х₂ = 3 ¹ = 3.

у₂ = 3 ^1/2 = √3.