Question

solve the following equation for x: \\(\ln(x + 9) + \ln(x + 3) = \ln(7)\\)
answer 2 points
separate multiple answers with a comma.
\\(x = \\)

Explanation:

Step1: Apply logarithm product rule

Using the property \(\ln a+\ln b = \ln(ab)\), we can rewrite the left - hand side of the equation \(\ln(x + 9)+\ln(x + 3)=\ln(7)\) as \(\ln[(x + 9)(x + 3)]=\ln(7)\).

Step2: Use one - to - one property of logarithms

If \(\ln m=\ln n\), then \(m = n\) (for \(m>0\) and \(n>0\)). So we have \((x + 9)(x + 3)=7\).

Step3: Expand the left - hand side

Expand \((x + 9)(x + 3)\) using the FOIL method: \(x^{2}+3x+9x + 27=7\).
Simplify the equation to get a quadratic equation: \(x^{2}+12x+27 - 7=0\), which is \(x^{2}+12x + 20=0\).

Step4: Solve the quadratic equation

Factor the quadratic equation \(x^{2}+12x + 20=0\). We need two numbers that multiply to \(20\) and add up to \(12\). The numbers are \(10\) and \(2\). So the factored form is \((x + 10)(x+2)=0\).
Set each factor equal to zero: \(x+10 = 0\) or \(x + 2=0\). This gives us \(x=-10\) or \(x=-2\).

Step5: Check the domain of the logarithm

For the original equation \(\ln(x + 9)+\ln(x + 3)=\ln(7)\), the arguments of the logarithms must be positive.

• For \(x=-10\): \(x + 9=-10 + 9=-1<0\) and \(x + 3=-10+3=-7<0\), so \(x=-10\) is not in the domain.
• For \(x=-2\): \(x + 9=-2 + 9 = 7>0\) and \(x + 3=-2+3 = 1>0\), so \(x=-2\) is in the domain.

Answer:

\(-2\)