Question 971225: Solve the following logarithmic equation
log2(x-4)=4-log2(x+2) Found 2 solutions by addingup, MathTherapy:Answer by addingup(3677) (Show Source):
You can put this solution on YOUR website! (log(x-4))/(log(2)) = 4-(log(x+2))/(log(2))
-4+(log(x-4))/(log(2))+(log(x+2))/(log(2)) = 0
-(4 log(2)-log(x-4)-log(x+2))/(log(2)) = 0
Multiply both sides by -log(2):
4 log(2)-log(x-4)-log(x+2) = 0
4 log(2)-log(x-4)-log(x+2) = log(16)+log(1/(x-4))+log(1/(x+2)) = log(16/((x-4) (x+2))):
log(16/((x-4) (x+2))) = 0
Take exp of both sides:
16/((x-4) (x+2)) = 1
Take the reciprocal of both sides:
1/16 (x-4) (x+2) = 1
Multiply both sides by 16:
(x-4) (x+2) = 16
Expand out terms on left:
x^2-2 x-8 = 16
Add 8 to both sides:
x^2-2 x = 24
Add 1 to both sides:
x^2-2 x+1 = 25
Write the left hand side as a square:
(x-1)^2 = 25
Take the square root of both sides:
x-1 = 5 or x-1 = -5
Add 1 to both sides:
x = 6 or x-1 = -5
Add 1 to both sides:
x = 6 or x = -4
(log(x-4))/(log(2)) => (log(-4-4))/(log(2)) = (i pi+log(8))/(log(2)) ~~ 3.+4.53236 i
4-(log(x+2))/(log(2)) => 4-(log(2-4))/(log(2)) = (-i pi+3 log(2))/(log(2)) ~~ 3.-4.53236 i:
So this solution is incorrect
(log(x-4))/(log(2)) => (log(6-4))/(log(2)) = 1
4-(log(x+2))/(log(2)) => 4-(log(2+6))/(log(2)) = 1:
So this solution is correct
Answer: x = 6
