B. The solution set is the set of real numbers, and the solution is x = 3, which satisfies the original logarithmic equation log₃(x³ - 18) = 2.
Let's solve the logarithmic equation log₃(x³ - 18) = 2 step by step:
1. Start with the given equation:
log₃(x³ - 18) = 2
2. Use the definition of logarithms to rewrite the equation in exponential form:
3² = x³ - 18
3. Simplify the right side:
9 = x³ - 18
4. Add 18 to both sides of the equation:
x³ = 27
5. Take the cube root of both sides to solve for x:
x = 3
Now, let's analyze the solution:
The solution x = 3 satisfies the original logarithmic equation, making it a valid solution. Therefore, the correct answer is:
B. The solution set is the set of real numbers.
To solve the given logarithmic equation log₃(x³ - 18) = 2, we first rewrite it in exponential form using the definition of logarithms. This gives us 3² = x³ - 18. Simplifying further, we get 9 = x³ - 18. By adding 18 to both sides, we isolate x³ on one side, resulting in x³ = 27. Finally, taking the cube root of both sides gives us the solution x = 3.
The solution set is not empty, and the value x = 3 satisfies the original logarithmic equation. Since the cube root of 27 is indeed 3, the solution is valid. Therefore, the correct answer is that the solution set is the set of real numbers, making option B the correct choice.