We are solving the equation log base 3 of x plus log base 3 of x minus 3 equals 2, with the condition that x is greater than 3. Let's first understand why this domain restriction is necessary.
Let's review key logarithm properties. The product rule states that the sum of two logarithms with the same base equals the logarithm of their product. For example, log base 3 of 9 plus log base 3 of 3 equals log base 3 of 27. The definition tells us that if log base a of x equals y, then a to the power of y equals x.
Now we apply the product rule to combine the logarithms. Log base 3 of x plus log base 3 of x minus 3 equals log base 3 of x times x minus 3. This simplifies to log base 3 of x squared minus 3x. So our equation becomes log base 3 of x squared minus 3x equals 2.
Using the definition of logarithms, if log base 3 of x squared minus 3x equals 2, then 3 to the power of 2 equals x squared minus 3x. This simplifies to 9 equals x squared minus 3x. Rearranging terms, we get the quadratic equation x squared minus 3x minus 9 equals 0.
We apply the quadratic formula to solve x squared minus 3x minus 9 equals 0. With a equals 1, b equals negative 3, and c equals negative 9, we calculate the discriminant as b squared minus 4ac, which is 9 plus 36 equals 45. The square root of 45 simplifies to 3 root 5. This gives us two solutions: x equals 3 plus 3 root 5 all over 2, and x equals 3 minus 3 root 5 all over 2.
Let's evaluate our solutions numerically. X sub 1 equals 3 plus 3 root 5 all over 2, which is approximately 4.854. X sub 2 equals 3 minus 3 root 5 all over 2, which is approximately negative 1.854. Since our domain requires x to be greater than 3, only the positive solution x sub 1 is valid. The negative solution is extraneous and must be rejected.
Let's verify our solution by substituting x equals 3 plus 3 root 5 all over 2 back into the original equation. We calculate x minus 3 and then the product x times x minus 3. Through algebraic manipulation, we find this product equals 9. Therefore, the left side of our equation becomes log base 3 of 9, which equals 2 since 3 squared equals 9. This confirms our solution is correct.