Solve for x natural log of x+ natural log of x+1=4

Problem

ln(x)+ln(x+1)=4

Solution

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

ln(x*(x+1))=4

2. Rewrite in exponential form by applying the base e to both sides to eliminate the natural logarithm.

x*(x+1)=e4

3. Distribute the x to form a quadratic equation.

x2+x=e4

4. Set the equation to zero by subtracting e4 from both sides.

x2+x−e4=0

5. Apply the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a) where a=1 b=1 and c=−e4

x=(−1±√(,1−4*(1)*(−e4)))/(2*(1))

6. Simplify the expression under the square root.

x=(−1±√(,1+4*e4))/2

7. Check for extraneous solutions by ensuring the argument of the original logarithms is positive (x>0. Since √(,1+4*e4)>1 the negative root would result in x<0

x=(−1+√(,1+4*e4))/2

Final Answer

ln(x)+ln(x+1)=4⇒x=(−1+√(,1+4*e4))/2

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