Solve for x

Problem

ln(1−x)+ln(2−x)=ln(x+14)

Solution

1. Identify the domain of the logarithmic functions by ensuring the arguments are positive: 1−x>0 2−x>0 and x+14>0

x<1

x<2

x>−14

Domain: −14<x<1

2. Apply the product rule for logarithms, ln(a)+ln(b)=ln(a*b) to the left side of the equation.

ln((1−x)*(2−x))=ln(x+14)

3. Exponentiate both sides to remove the natural logarithms, using the property eln(u)=u

(1−x)*(2−x)=x+14

4. Expand the product on the left side using the FOIL method.

2−x−2*x+x2=x+14

x2−3*x+2=x+14

5. Rearrange into a quadratic equation by subtracting x and 14 from both sides to set the equation to zero.

x2−4*x−12=0

6. Factor the quadratic expression by finding two numbers that multiply to −12 and add to −4

(x−6)*(x+2)=0

7. Solve for x by setting each factor to zero.

x=6

x=−2

8. Check against the domain established in step 1. Since x=6 is not less than 1 it is an extraneous solution.

x=−2

Final Answer

ln(1−x)+ln(2−x)=ln(x+14)⇒x=−2

Want more problems? Check here!