Solve for $x$.
\[
\ln 2+\ln (x+8)=\ln 12
\]
Answer
We can check the solution by substitifying x=-2 into the original equation: \(\ln 2+\ln (-2+8)=\ln 12\), which simplifies to: \(\ln 2+\ln 6=\ln 12\). Using the property of logarithms that the sum of two logarithms is the logarithm of the product, we get: \(\ln (2*6)=\ln 12\), which simplifies to: \(\ln 12=\ln 12\). Since the two sides of the equation are equal, the solution x=-2 is correct.
Step 1 ：The given equation is: \(\ln 2+\ln (x+8)=\ln 12\)
Step 2 ：Using the property of logarithms that the sum of two logarithms with the same base is equal to the logarithm of the product of the numbers, we can rewrite the left side of the equation as: \(\ln (2*(x+8))=\ln 12\), which simplifies to: \(\ln (2x+16)=\ln 12\)
Step 3 ：Using the property of logarithms that if two logarithms with the same base are equal, then their arguments are also equal, we can set the arguments equal to each other: \(2x+16=12\)
Step 4 ：Solving for x, we subtract 16 from both sides of the equation: \(2x=12-16\), which simplifies to: \(2x=-4\). Dividing both sides by 2, we get: \(x=-4/2\)
Step 5 ：\(\boxed{x=-2}\)
Step 6 ：We can check the solution by substitifying x=-2 into the original equation: \(\ln 2+\ln (-2+8)=\ln 12\), which simplifies to: \(\ln 2+\ln 6=\ln 12\). Using the property of logarithms that the sum of two logarithms is the logarithm of the product, we get: \(\ln (2*6)=\ln 12\), which simplifies to: \(\ln 12=\ln 12\). Since the two sides of the equation are equal, the solution x=-2 is correct.
