1. The problem asks to sketch the graph of the function $$y = 4\ln(2x - 4)$$. 2. The natural logarithm function $$\ln(x)$$ is defined only for $$x > 0$$, so the domain of $$y = 4\ln(2x - 4)$$ is where $$2x - 4 > 0$$. 3. Solve the inequality for the domain: $$ 2x - 4 > 0 \\ 2x > 4 \\ x > 2 $$ So the domain is $$x > 2$$. 4. The function is a vertical stretch of the basic logarithm function by a factor of 4. 5. The vertical asymptote occurs where the argument of the logarithm is zero: $$ 2x - 4 = 0 \\ x = 2 $$ So the vertical asymptote is the line $$x = 2$$. 6. To sketch the graph, plot some points for $$x > 2$$: - At $$x=3$$: $$y = 4\ln(2(3) - 4) = 4\ln(6 - 4) = 4\ln(2) \approx 4 \times 0.693 = 2.772$$ - At $$x=4$$: $$y = 4\ln(2(4) - 4) = 4\ln(8 - 4) = 4\ln(4) \approx 4 \times 1.386 = 5.544$$ 7. The graph approaches negative infinity as $$x$$ approaches 2 from the right. 8. The graph increases slowly for larger $$x$$ values due to the logarithmic nature. Final answer: The graph of $$y = 4\ln(2x - 4)$$ has domain $$x > 2$$, vertical asymptote at $$x=2$$, and is a vertically stretched logarithmic curve increasing slowly to the right.