How do you graph y=ln x-1?

Graphing the function \( y = \ln x - 1 \) involves understanding the natural logarithm function and how transformations affect its graph. Let’s break this down step by step.

Understanding the Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is the inverse of the exponential function \( e^x \). The graph of \( y = \ln x \) has some key characteristics:

• It is defined only for \( x > 0 \).
• As \( x \) approaches 0 from the right, \( y \) approaches negative infinity.
• At \( x = 1 \), \( y = 0 \) because \( \ln(1) = 0 \).
• As \( x \) increases, \( y \) increases but at a decreasing rate.

Transforming the Graph

The equation \( y = \ln x - 1 \) represents a vertical shift of the basic logarithmic function. Specifically, subtracting 1 from \( \ln x \) shifts the entire graph down by 1 unit.

Steps to Graph \( y = \ln x - 1 \)

1. Start with the basic graph of \( y = \ln x \): Plot points for values of \( x \) such as 0.1, 0.5, 1, 2, and 5. You will notice that as \( x \) increases, \( y \) increases but at a slower pace.
2. Apply the vertical shift: For each point on the graph of \( y = \ln x \), subtract 1 from the \( y \)-coordinate. For example:
   • At \( x = 1 \), \( y = \ln(1) - 1 = 0 - 1 = -1 \).
   • At \( x = 2 \), \( y = \ln(2) - 1 \approx 0.693 - 1 \approx -0.307 \).
   • At \( x = 0.5 \), \( y = \ln(0.5) - 1 \approx -0.693 - 1 \approx -1.693 \).
3. Plot the transformed points: Mark the new coordinates on your graph. For instance, you would plot (1, -1), (2, -0.307), and (0.5, -1.693).
4. Draw the curve: Connect the points smoothly, keeping in mind the shape of the logarithmic function. The graph will approach the vertical line \( x = 0 \) but never touch it, continuing downwards as \( x \) approaches 0.

Key Features of the Graph

After plotting, you’ll notice several important features:

• The domain is \( x > 0 \).
• The range is all real numbers, as the graph can go down indefinitely.
• The graph crosses the line \( y = -1 \) at \( x = 1 \).
• It has a vertical asymptote at \( x = 0 \).

Visualizing the Graph

To visualize this better, imagine the original \( y = \ln x \) graph, which starts at negative infinity as \( x \) approaches 0 and crosses the x-axis at \( x = 1 \). Now, picture taking that entire graph and moving it down by one unit. This gives you the graph of \( y = \ln x - 1 \), which retains the same shape but is positioned lower on the y-axis.

By following these steps, you should be able to graph \( y = \ln x - 1 \) accurately and understand how transformations affect the logarithmic function. If you have any further questions or need clarification on any part of this process, feel free to ask!