Logarithmic Graph F25830

1. **State the problem:** We need to graph the function $$y = \log_2(x - 1) - 2$$ including its vertical asymptote and curve. 2. **Identify the vertical asymptote:** The logarithmic function $$\log_2(x - 1)$$ is defined only for $$x - 1 > 0$$, so $$x > 1$$. The vertical asymptote occurs where the argument of the logarithm is zero: $$x - 1 = 0 \implies x = 1$$. 3. **Graph the asymptote:** Draw a vertical dashed line at $$x = 1$$. 4. **Understand the transformations:** - The base function is $$y = \log_2 x$$. - The function is shifted right by 1 unit (inside the log: $$x - 1$$). - The function is shifted down by 2 units (outside the log: $$- 2$$). 5. **Plot key points:** - When $$x = 2$$, $$y = \log_2(2 - 1) - 2 = \log_2(1) - 2 = 0 - 2 = -2$$. - When $$x = 3$$, $$y = \log_2(3 - 1) - 2 = \log_2(2) - 2 = 1 - 2 = -1$$. - When $$x = 5$$, $$y = \log_2(5 - 1) - 2 = \log_2(4) - 2 = 2 - 2 = 0$$. 6. **Behavior of the curve:** - The curve lies to the right of the asymptote $$x=1$$. - It increases slowly as $$x$$ increases. - It passes through points (2, -2), (3, -1), and (5, 0). 7. **Summary:** - Vertical asymptote at $$x=1$$. - Curve shifted right 1 and down 2 from $$y=\log_2 x$$. - Curve lies to the right of the asymptote and increases slowly. **Final answer:** The graph consists of a vertical dashed line at $$x=1$$ and a logarithmic curve starting just to the right of $$x=1$$, passing through the points calculated above, increasing slowly upwards.