SOLUTION: Solve for x: 2 ln x - ln (x+1) = 3 Can someone please solve this problem step by step for me and explain how and why it is solved a certain way? Thank you!

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Solve for x: 2 ln x - ln (x+1) = 3 Can someone please solve this problem step by step for me and explain how and why it is solved a certain way? Thank you!      Log On


   

Question 1025485: Solve for x: 2 ln x - ln (x+1) = 3
Can someone please solve this problem step by step for me and explain how and why it is solved a certain way? Thank you!
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Solve for x: 2 ln x - ln (x+1) = 3
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ln(x^2) - ln(x+1) = 3
ln(x^2) = ln(x+1) + 3
ln(x^2) = ln(x+1) + ln(e^3)
ln(x^2) = ln(e^3*(x+1))
x^2 = e^3*x+1
x^2 - e^3x - 1 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-20.0855369231877x%2B-1+=+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=%28-20.0855369231877%29%5E2-4%2A1%2A-1=407.428793492736.

Discriminant d=407.428793492736 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--20.0855369231877%2B-sqrt%28+407.428793492736+%29%29%2F2%5Ca.




Quadratic expression 1x%5E2%2B-20.0855369231877x%2B-1 can be factored:
1x%5E2%2B-20.0855369231877x%2B-1+=+%28x-20.135201189791%29%2A%28x--0.0496642666032567%29
Again, the answer is: 20.135201189791, -0.0496642666032567. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-20.0855369231877%2Ax%2B-1+%29

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The negative solution is not allowed due to ln(x).
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email any questions about the method via the TY note.