SOLUTION: lnx=ln(2x-1)-ln(x-2)

Question 298804: lnx=ln(2x-1)-ln(x-2)
Found 2 solutions by user_dude2008, nerdybill:
Answer by user_dude2008(1862) (Show Source): You can put this solution on YOUR website!
ln(x)=ln(2x-1)-ln(x-2)

ln(x)=ln[(2x-1)/(x-2)]

x=(2x-1)/(x-2)

x(x-2)=2x-1

x^2-2x=2x-1

x^2-2x-2x+1=0

x^2-4x+1=0

Answer: x = 2+sqrt(3) or x = 2-sqrt(3)
Answer by nerdybill(7384) (Show Source): You can put this solution on YOUR website!
lnx=ln(2x-1)-ln(x-2)
lnx=ln(2x-1)/(x-2)
x=(2x-1)/(x-2)
x(x-2)=(2x-1)
x^2-2x=2x-1
x^2-4x=-1
x^2-4x+1=0
Since we can't factor, we must resort to the quadratic formula. Doing so yields:
x = {3.732, 0.268}
We can toss out 0.268 because if we use it in the original equation it produces a ln of a negative number -- toss it out, it is an extraneous solution. This leaves:
x = 3.732
.
Details of quadratic formula to follow:

Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation (in our case ) has the following solutons:

For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=12 is greater than zero. That means that there are two solutions: .

Quadratic expression can be factored:

Again, the answer is: 3.73205080756888, 0.267949192431123. Here's your graph:

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