Solve for x natural log of x-8+ natural log of x+9=1

Problem

ln(x−8)+ln(x+9)=1

Solution

1. Apply the product rule for logarithms, which states that ln(a)+ln(b)=ln(a*b)

ln((x−8)*(x+9))=1

2. Rewrite in exponential form by applying the base e to both sides of the equation.

(x−8)*(x+9)=e1

3. Expand the product on the left side of the equation.

x2+9*x−8*x−72=e

x2+x−72=e

4. Set the equation to zero to form a standard quadratic equation.

x2+x−(72+e)=0

5. Apply the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a) where a=1 b=1 and c=−(72+e)

x=(−1±√(,1−4*(1)*(−(72+e))))/(2*(1))

x=(−1±√(,1+4*(72+e)))/2

x=(−1±√(,1+288+4*e))/2

x=(−1±√(,289+4*e))/2

6. Check for extraneous solutions by ensuring the arguments of the original logarithms are positive (x−8>0 and x+9>0. This requires x>8

x=(−1+√(,289+4*e))/2

Final Answer

ln(x−8)+ln(x+9)=1⇒x=(−1+√(,289+4*e))/2

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