Solve for x

Problem

ln(x−6)+ln(x−5)=ln(16−2*x)

Solution

1. Identify the domain of the logarithmic functions by ensuring all arguments are positive: x−6>0 x−5>0 and 16−2*x>0

x>6

x>5

x<8

Domain: *6<x<8

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

ln((x−6)*(x−5))=ln(16−2*x)

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

(x−6)*(x−5)=16−2*x

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

x2−5*x−6*x+30=16−2*x

x2−11*x+30=16−2*x

5. Rearrange into a quadratic equation by moving all terms to one side.

x2−9*x+14=0

6. Factor the quadratic expression.

(x−7)*(x−2)=0

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

x=7

x=2

8. Check the solutions against the domain 6<x<8 The value x=2 is extraneous because it results in taking the logarithm of a negative number.

x=7

Final Answer

ln(x−6)+ln(x−5)=ln(16−2*x)⇒x=7

Want more problems? Check here!