SOLUTION: I need help on a logarithm problem. I need to Solve for X. 2ln(x)-ln(x+8/3)= ln(6)

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: I need help on a logarithm problem. I need to Solve for X. 2ln(x)-ln(x+8/3)= ln(6)      Log On


   

Question 306865: I need help on a logarithm problem.
I need to Solve for X.
2ln(x)-ln(x+8/3)= ln(6)
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
2ln(x)-ln((x+8)/3)= ln(6)
ln(x^2)-ln((x+8)/3)= ln(6)
ln(x^2)/((x+8)/3)= ln(6)
(x^2)/((x+8)/3)= 6
(x^2) = 6((x+8)/3)
(x^2) = 2(x+8)
(x^2) = 2x+16
x^2-2x-16 = 0
.
Solve by applying the quadratic formula. Doing so yields:
x = {5.123, -3.123}
Toss out the negative solution (extraneous) leaving:
x = 5.123
.
Details of quadratic formula follows:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-2x%2B-16+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-2%29%5E2-4%2A1%2A-16=68.

Discriminant d=68 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--2%2B-sqrt%28+68+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-2%29%2Bsqrt%28+68+%29%29%2F2%5C1+=+5.12310562561766
x%5B2%5D+=+%28-%28-2%29-sqrt%28+68+%29%29%2F2%5C1+=+-3.12310562561766

Quadratic expression 1x%5E2%2B-2x%2B-16 can be factored:
1x%5E2%2B-2x%2B-16+=+1%28x-5.12310562561766%29%2A%28x--3.12310562561766%29
Again, the answer is: 5.12310562561766, -3.12310562561766. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-2%2Ax%2B-16+%29