Solve for x natural log of x+ natural log of x+10=3

Problem

ln(x)+ln(x+10)=3

Solution

1. Apply the product rule for logarithms to combine the terms on the left side into a single logarithm.

ln(x*(x+10))=3

2. Rewrite in exponential form by using the base e to eliminate the natural logarithm.

x*(x+10)=e3

3. Distribute the variable to form a quadratic equation.

x2+10*x=e3

4. Set the equation to zero by subtracting e3 from both sides.

x2+10*x−e3=0

5. Apply the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a) where a=1 b=10 and c=−e3

x=(−10±√(,10−4*(1)*(−e3)))/(2*(1))

6. Simplify the expression under the square root.

x=(−10±√(,100+4*e3))/2

7. Factor out a 4 from the square root to simplify the fraction.

x=(−10±2√(,25+e3))/2

8. Divide by 2 to find the potential solutions.

x=−5±√(,25+e3)

9. Check for extraneous solutions by ensuring the argument of the original logarithms is positive. Since x must be greater than 0 we discard the negative root.

x=−5+√(,25+e3)

Final Answer

ln(x)+ln(x+10)=3⇒x=−5+√(,25+e3)

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