Solve for x

Problem

ln(x−2)−ln(x+2)=ln(x−1)−ln(2*x+1)

Solution

1. Apply the quotient rule for logarithms, ln(a)−ln(b)=ln(a/b) to both sides of the equation.

ln((x−2)/(x+2))=ln((x−1)/(2*x+1))

2. Exponentiate both sides to remove the natural logarithms, using the property that if ln(A)=ln(B) then A=B

(x−2)/(x+2)=(x−1)/(2*x+1)

3. Cross-multiply to eliminate the fractions and create a polynomial equation.

(x−2)*(2*x+1)=(x−1)*(x+2)

4. Expand both sides of the equation using the FOIL method.

2*x2+x−4*x−2=x2+2*x−x−2

2*x2−3*x−2=x2+x−2

5. Rearrange the equation into standard quadratic form by subtracting x2+x−2 from both sides.

x2−4*x=0

6. Factor the quadratic expression to find the potential solutions for x

x*(x−4)=0

x=0* or *x=4

7. Check for extraneous solutions by ensuring the arguments of the original logarithms are positive. For x=0 ln(0−2) is undefined. For x=4 all arguments (4−2,4+2,4−1,8+1) are positive.

x=4

Final Answer

ln(x−2)−ln(x+2)=ln(x−1)−ln(2*x+1)⇒x=4

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