Solve for x

Problem

ln(x)+ln(x+6)=1/2*ln(9)

Solution

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

ln(x*(x+6))=1/2*ln(9)

2. Apply the power rule for logarithms on the right side, which states n*ln(a)=ln(an)

ln(x2+6*x)=ln(9(1/2))

3. Simplify the constant on the right side by calculating the square root of 9.

ln(x2+6*x)=ln(3)

4. Exponentiate both sides to remove the natural logarithms, setting the arguments equal to each other.

x2+6*x=3

5. Rearrange into a quadratic equation by subtracting 3 from both sides to set the equation to zero.

x2+6*x−3=0

6. Apply the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a) where a=1 b=6 and c=−3

x=(−6±√(,6−4*(1)*(−3)))/(2*(1))

7. Simplify the discriminant and the resulting expression.

x=(−6±√(,36+12))/2

x=(−6±√(,48))/2

x=(−6±4√(,3))/2

x=−3±2√(,3)

8. Check for extraneous solutions by ensuring the arguments of the original logarithms are positive. Since x>0 we must reject x=−3−2√(,3)

x=−3+2√(,3)

Final Answer

ln(x)+ln(x+6)=1/2*ln(9)⇒x=−3+2√(,3)

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