Final Answer:
y = log₃(0.5x - 6) is the equation for a logarithmic function with a base of 3 that is half as wide as the parent function and is shifted 6 units right. option B is correct.
Step-by-step explanation:
In a logarithmic function, the general form is y = logₐ(bx - h) + k, where 'a' is the base, 'b' is the stretch or compression factor, 'h' is the horizontal shift, and 'k' is the vertical shift. For the given question, we know the base is 3, the function is half as wide as the parent function (stretch factor), and it is shifted 6 units to the right (horizontal shift).
Let's break down the components of the final answer:
1. Base and Stretch/Compression Factor:
The base is 3, and since the function is half as wide as the parent function, the stretch or compression factor 'b' is 0.5. This gives us log₃(0.5x - h).
2. Horizontal Shift:
The function is shifted 6 units to the right, so the horizontal shift 'h' is -6 (opposite direction of the shift). This gives us log₃(0.5x - 6).
Therefore, combining these elements, the correct answer is B) y = log₃(0.5x - 6). This equation represents a logarithmic function with a base of 3, half as wide as the parent function, and shifted 6 units to the right.
This conclusion aligns with the principles of logarithmic functions, ensuring that the given specifications are correctly incorporated into the equation. By understanding the transformations associated with logarithmic functions, we can systematically deduce the appropriate form of the equation.