How do you graph #F(x)=ln(x-3)#?

Question

Caleb Alexander · Accepted Answer

y = F(x). Use the inverse relation #x = 3 + e^y. x>=3#. Graph meets x-axis at (4, 0). x =3 is the vertical asymptote, in the negative y-direction. Graph is from #(3+, -oo) to (oo, oo)# through (4, 0).

Plot points (3.25, ln 0.25), (3.5, ln 0.5), (4, 0), (4.5, ln 1.5), (5, ln 2),...to see how it rises from #(3+, -oo) to (oo, oo)#. through these points.

Ariana Alvarez · Answer

undefined To graph $ F(x) = \ln(x - 3) $, you can follow these steps:

1. Determine the domain of the function. Since the natural logarithm function $ \ln(x) $ is defined only for positive real numbers, in this case, $ x - 3 > 0 $, so $ x > 3 $.

2. Find the vertical asymptote. The vertical asymptote occurs where the argument of the logarithm is zero, so set $ x - 3 = 0 $, which gives $ x = 3 $.

3. Determine the behavior of the function as $ x $ approaches the asymptote. As $ x $ approaches 3 from the right side, $ \ln(x - 3) $ goes to negative infinity, and as $ x $ becomes larger than 3, $ \ln(x - 3) $ increases without bound.

4. Choose some points to plot. Pick a few values of $ x $ greater than 3, calculate the corresponding $ y $-values using $ \ln(x - 3) $, and plot these points.

5. Sketch the graph. Based on the behavior near the asymptote and the points you've plotted, sketch the curve. It should approach the vertical asymptote at $ x = 3 $, increase slowly at first, then more steeply as $ x $ increases.

6. Label the axes and any important points. Label the vertical asymptote at $ x = 3 $, and if needed, indicate the direction of the curve as it approaches the asymptote.

7. Your graph should resemble a logarithmic curve, approaching the vertical asymptote at $ x = 3 $ and increasing without bound as $ x $ increases.