log₁₅₀(5) = 1 / log₅(150)
log₅(30) : (1 / log₅(150)) = log₅(30) * log₅(150) = log₅(5*6) * log₅(6*25) =
= ( log₅(5)+log₅(6) ) * ( log₅(25)+log₅(6) ) = ( 1+log₅(6) ) * ( 2+log₅(6) )
аналогично:
log₅(750) : (1 / log₅(6)) = log₅(750) * log₅(6) = log₅(6*125) * log₅(6) =
= ( log₅(125)+log₅(6) ) * log₅(6) = ( 3+log₅(6) ) * log₅(6)
осталось вычесть... удобно обозначить x = log₅(6)
( 1+x ) * ( 2+x ) - ( 3+x ) * x = 2+x+2x+x² - 3x-x² = 2
log₂(70) : (1 / log₂(280)) = log₂(70) * log₂(280) = log₂(35*2) * log₂(35*8) =
= ( log₂(2)+log₂(35) ) * ( log₂(8)+log₂(35) ) = ( 1+log₂(35) ) * ( 3+log₂(35) )
аналогично:
log₂(560) : (1 / log₂(35)) = log₂(560) * log₂(35) = log₂(16*35) * log₂(35) =
= ( log₂(16)+log₂(35) ) * log₂(35) = ( 4+log₂(35) ) * log₂(35)
осталось вычесть... удобно обозначить x = log₂(35)
( 1+x ) * ( 3+x ) - ( 4+x ) * x = 3+x+3x+x² - 4x-x² = 3
2X^2 - X * ( 2X - 2 ) = 6
2X^2 - 2X^2 + 2X = 6
2X = 6
X = 3
Y = 6 - 2 = 4
ОТВЕТ ( 3 ; 4 )
( X + 2 )*( Y + 1 ) = 12
X + 2Y = 6 ; X = 6 - 2Y
( 6 - 2Y + 2 )*( Y + 1 ) = 12
( 8 - 2Y )*( Y + 1 ) = 12
8Y + 8 - 2Y^2 - 2Y = 12
- 2Y^2 + 6Y - 4 = 0
- 2 * ( Y^2 - 3Y + 2 ) = 0
D = 9 - 8 = 1 ; √ D = 1
Y1 = ( 3 + 1 ) : 2 = 2
Y2 = ( 3 - 1 ) : 2 = 1
X1 = 6 - 4 = 2
X2 = 6 - 2 = 4
ОТВЕТ ( 2 ; 2 ) ; ( 4 ; 1 )
X^2 + Y^2 = 10
XY = - 3
X = ( - 3 / Y ) ; X^2 = 9 / Y^2
( 9 / Y^2 ) + Y^2 = 10
( 9 + Y^4 ) / Y^2 = 10 ( Y ≠ 0 )
9 + Y^4 = 10Y^2
Y^4 - 10Y^2 + 9 = 0
Y^2 = A ; A > 0
A^2 - 10A + 9 = 0
D = 100 - 36 = 64 ; √ D = 8
A1 = ( 10 + 8 ) : 2 = 9
A2 = ( 10 - 8 ) : 2 = 1
Y^2 = 9 ===> Y (1 /2 ) = ( + / - ) 3
Y^2 = 1 ===> Y ( 3/4 ) = ( +/ - ) 1
X^2 = 9 / Y^2
X^2 = 9 / 9 = 1 ===> X ( 1/2 ) = ( + / - ) 1
X^2 = 9 / 1 = 9 ===> X ( 3/4 ) = ( + / - ) 3
ОТВЕТ ( 1 ; 3 ); ( - 1 ; - 3 ); ( 3 ; 1 ) ; ( - 3 ; - 1 )