Graph of y = ln(x)

Definition

The graph of y = ln(x) represents the natural logarithm function, which is the inverse of the exponential function with base e. This graph is defined for all positive values of x, approaching negative infinity as x approaches zero from the right, and increasing without bound as x increases. Key features of this graph include its unique shape and critical points that reflect important properties of logarithmic functions.

5 Must Know Facts For Your Next Test

1. The graph of y = ln(x) passes through the point (1, 0) because ln(1) equals 0.
2. As x approaches 0 from the right, the value of ln(x) approaches negative infinity, indicating a vertical asymptote at x = 0.
3. The slope of the graph at any point is determined by its derivative, which is 1/x; this means it becomes less steep as x increases.
4. The function is continuous and monotonically increasing, meaning it always rises as you move to the right along the x-axis.
5. The second derivative test shows that the function is concave down for all x > 0, which reflects its decreasing rate of increase.

Review Questions

• How does the behavior of the graph of y = ln(x) near its asymptote at x = 0 reflect its mathematical properties?
  • As x approaches 0 from the right, the graph of y = ln(x) plunges downwards towards negative infinity, indicating that ln(x) is undefined for non-positive values. This behavior highlights the function's domain being limited to positive real numbers only. The vertical asymptote at x = 0 serves as a boundary beyond which the logarithm does not exist, emphasizing the importance of positive inputs in logarithmic functions.
• Discuss how the derivative of y = ln(x) influences its graph's slope and overall shape.
  • The derivative of y = ln(x), which is 1/x, indicates that as x increases, the slope of the graph becomes less steep. This means that while the function continues to rise indefinitely, it does so at a decreasing rate. Consequently, the graph becomes flatter as you move to the right. This relationship between the function and its derivative illustrates how logarithmic functions grow more slowly compared to exponential functions.
• Evaluate how understanding the properties of y = ln(x) can enhance your ability to analyze other logarithmic functions.
  • Understanding the properties of y = ln(x) provides a foundation for analyzing other logarithmic functions because it reveals key characteristics such as domain restrictions and behavior at asymptotes. By recognizing that logarithmic functions are inversely related to exponential functions, you can anticipate transformations such as horizontal shifts or vertical stretches. This knowledge allows you to make predictions about similar graphs and their respective behaviors in terms of continuity and monotonicity.

Related terms

Natural Exponential Function: The function defined as y = e^x, which serves as the inverse to the natural logarithm function.

Domain:

The set of all possible input values (x-values) for the function, which for ln(x) is (0, ∞).

Asymptote:

A line that the graph approaches but never touches; for y = ln(x), the vertical line x = 0 serves as a vertical asymptote.