Graph the function below by moving the key points. \[ f(x)=2 \log _{3}(x) \]

We want to graph   f(x) = 2·log₃(x) by “moving” key points. One good strategy is to start with the basic logarithm function g(x) = log₃(x) and then apply the transformation that multiplies the output by 2 (a vertical stretch). Step 1. Identify the basic graph of g(x) = log₃(x):  • Domain: x > 0.  • Vertical asymptote at x = 0.  • The point (1, 0) is on the graph because log₃(1) = 0.  • Other easy-to-calculate points:   – When x = 3, log₃(3) = 1, so the point (3, 1) is on g(x).   – When x = 1/3, log₃(1/3) = –1, so (1/3, –1) is on g(x). Step 2. Apply the transformation f(x) = 2·log₃(x)  • This multiplies all y-values by 2. In other words, we “stretch” the graph vertically by a factor of 2.  • The new key points become:   – (1, 0) stays the same because 2·0 = 0.   – (3, 1) moves to (3, 2) because 2·1 = 2.   – (1/3, –1) moves to (1/3, –2) because 2·(–1) = –2. Step 3. Describe the new graph:  • Domain: Still x > 0.  • Vertical asymptote: Still the line x = 0.  • Behavior:   – As x → 0⁺, f(x) → –∞.   – f(1) = 2·log₃(1) = 0.   – As x increases further, f(x) increases but slowly (logarithmic growth); for instance, f(9) = 2·log₃(9) = 2·2 = 4.  • The graph curves upward slowly, with the same “shape” as log₃(x) but with values doubled. Step 4. Sketching the graph:  1. Draw a vertical dashed line at x = 0 to indicate the asymptote.  2. Plot the key points:   – (1, 0)   – (3, 2)   – (1/3, –2)  3. Sketch a smooth curve passing through these points that approaches –∞ as x → 0⁺ and rises gradually as x increases. Your final graph of f(x) = 2·log₃(x) shows a logarithmic curve with a vertical stretch relative to the basic log₃(x) graph.