We are solving the equation log base 3 of x plus log base 3 of x minus 2 equals 2. The domain restriction x greater than 2 ensures that both arguments of the logarithms are positive, which is necessary for the logarithms to be defined.
To solve our equation, we'll use two key logarithm properties. First, the sum of two logarithms with the same base equals the logarithm of their product. Second, if the logarithm of b base a equals c, then a to the power of c equals b. These properties will help us simplify and solve our equation.
We apply the logarithm addition property to combine the two logarithms on the left side of our equation. This transforms log base 3 of x plus log base 3 of x minus 2 into log base 3 of x times x minus 2, which equals 2.
Now we convert the logarithmic equation to its exponential form. If log base 3 of x times x minus 2 equals 2, then 3 squared equals x times x minus 2. Simplifying, we get 9 equals x squared minus 2x.
We rearrange the equation to standard quadratic form by moving all terms to one side. This gives us x squared minus 2x minus 9 equals 0. Identifying the coefficients, we have a equals 1, b equals negative 2, and c equals negative 9. We're now ready to apply the quadratic formula.
Applying the quadratic formula with a equals 1, b equals negative 2, and c equals negative 9, we get x equals 2 plus or minus square root of 40 all over 2. Simplifying further, this becomes x equals 1 plus or minus square root of 10. Numerically, these solutions are approximately 4.16 and negative 2.16.
We must check which of our solutions satisfy the original domain restriction x greater than 2. Our first solution, approximately 4.16, is greater than 2, so it's valid. Our second solution, approximately negative 2.16, is less than 2, so it's invalid and must be rejected.
Let's verify our solution by substituting x equals 1 plus square root of 10 back into the original equation. We find that x minus 2 equals square root of 10 minus 1. When we compute the sum of the logarithms, it simplifies to log base 3 of 9, which equals 2. This confirms our solution is correct.