Solve for x natural log of x^2-x = natural log of 6

Problem

ln(x^2−x)=ln(6)

Solution

1. Apply the one-to-one property of logarithms, which states that if ln(a)=ln(b) then a=b

x^2−x=6

2. Rearrange the equation into standard quadratic form a*x^2+b*x+c=0 by subtracting 6 from both sides.

x^2−x−6=0

3. Factor the quadratic expression by finding two numbers that multiply to −6 and add to −1

(x−3)*(x+2)=0

4. Apply the zero product property to find the potential solutions for x

x−3=0⇒x=3

x+2=0⇒x=−2

5. Verify the solutions by checking if they result in positive arguments for the original logarithms.
   For x=3 3^2−3=6>0 (Valid)
   For x=−2 (−2)^2−(−2)=6>0 (Valid)

Final Answer

ln(x^2−x)=ln(6)⇒x=3,−2

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