Question
Solve simultaneously log (x-1)+2logy=2log3 and logx+logy=log6
Answers
log ( x -1 ) + 2 log y = 2 log 3
log ( x -1 ) = 2 log 3 - 2 log y
log ( x -1 ) = 2 ( log 3 - log y )
log ( x -1 ) = 2 log ( 3 / y )
log ( x -1 ) = log [ ( 3 / y )² ]
x - 1 = ( 3 / y )²
x - 1 = 9 / y²
x = 9 / y² + 1
log x + log y = log 6
log ( x ∙ y ) = log 6
x ∙ y = 6
x = 6 / y
Use fact:
x = x
9 / y² + 1 = 6 / y
Multiply both sides by y²
9 + y² = 6 y
Subtract 6 y to both sides
9 + y² - 6 y = 6 y - 6 y
y² - 6 y + 9 = 0
Factor the left hand side
( y - 3 )² = 0
y - 3 = √0
y - 3 = 0
add 3 to both sides
y - 3 + 3 = 0 + 3
y = 3
x = 6 / y
x = 6 / 3
x = 2
Solution x = 2 , y = 3
Proof:
log ( x -1 ) + 2 log y = 2 log 3
log ( 2 -1 ) + 2 log 3 = 2 log 3
log 1 + 2 log 3 = 2 log 3
0 + 2 log 3 = 2 log 3
2 log 3 = 2 log 3
log x + log y = log 6
log 2 + log 3 = log 6
log ( 2 ∙ 3 ) = log 6
log 6 = log 6
log ( x -1 ) = 2 log 3 - 2 log y
log ( x -1 ) = 2 ( log 3 - log y )
log ( x -1 ) = 2 log ( 3 / y )
log ( x -1 ) = log [ ( 3 / y )² ]
x - 1 = ( 3 / y )²
x - 1 = 9 / y²
x = 9 / y² + 1
log x + log y = log 6
log ( x ∙ y ) = log 6
x ∙ y = 6
x = 6 / y
Use fact:
x = x
9 / y² + 1 = 6 / y
Multiply both sides by y²
9 + y² = 6 y
Subtract 6 y to both sides
9 + y² - 6 y = 6 y - 6 y
y² - 6 y + 9 = 0
Factor the left hand side
( y - 3 )² = 0
y - 3 = √0
y - 3 = 0
add 3 to both sides
y - 3 + 3 = 0 + 3
y = 3
x = 6 / y
x = 6 / 3
x = 2
Solution x = 2 , y = 3
Proof:
log ( x -1 ) + 2 log y = 2 log 3
log ( 2 -1 ) + 2 log 3 = 2 log 3
log 1 + 2 log 3 = 2 log 3
0 + 2 log 3 = 2 log 3
2 log 3 = 2 log 3
log x + log y = log 6
log 2 + log 3 = log 6
log ( 2 ∙ 3 ) = log 6
log 6 = log 6