Final answer:
To solve the logarithmic equation, apply the product rule of logarithms to combine terms, convert to exponential form, and solve the quadratic equation, checking for extraneous solutions and rounding the final answer to the nearest hundredth.
Step-by-step explanation:
The student is asking to solve the logarithmic equation log₃(x-9) +log₃(x-3) = 2. We can use logarithm properties to combine the terms on the left side of the equation since they have the same base.
First, apply the product rule of logarithms, which states that log₃(a)+log₃(b) = log₃(a*b). This gives us log₃((x-9)(x-3)) = 2.
To solve for x, we convert the logarithmic equation to its exponential form: 3² = (x-9)(x-3), which simplifies to 9 = (x-9)(x-3). We then expand and solve the resulting quadratic equation.
Next, we need to verify our solution(s) to ensure they do not create a negative number or zero inside the logarithm, which would result in extraneous solutions.
Lastly, we will round our final valid solution(s) to the nearest hundredth, as requested.