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Chapter 7: Problem 19

Use a graphing utility to graph the inequality. $$y<\ln x$$

Short Answer

Expert verified
The graph of the inequality \(y < \ln x\) is finished. It's the region below the curve \(y = \ln x\), excluding the line itself, which should be represented as a dashed line.

Step by step solution

01

- Understanding the logarithm function

The base of natural logarithm \(\ln x\) is the Euler's number \(e\), i.e., \(\ln x = \log_{e}x\). It has a logarithm curve that increases steadily as \(x\) goes from \(0\) to \(\infty\) and \(y\) goes from \(-\infty\) to \(\infty\). The function is undefined for \(x 鈮� 0\). Now it is necessary to transform this understanding to a graph.
02

- Graphing the logarithm function

Use the graphing utility to plot \(y = \ln x\). The curve should start from \(x = 1\) (where \(y = 0\)), then decrease towards negative infinity as \(x\) approaches \(0\) from the right, and increases towards positive infinity as \(x\) increases.
03

- Graphing the inequality

The inequality \(y < \ln x\) means we are interested in all the \(y\) values that are less than the \(y\) values of \(\ln x\). To represent this on the graph, shade all the region below the curve of \(y = \ln x\). Do not include the line itself, because the inequality is strictly less than, not less than or equal to. Hence, the graph of the function should be a dashed line.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Logarithm Function
The natural logarithm function, denoted as \(\ln x\), is a fundamental mathematical concept. It's important to remember that the base of the natural logarithm is the Euler's number, \(e\), approximately equal to 2.71828. Here's what you need to know about the \(\ln x\) function:
  • It is only defined for \(x > 0\).
  • The function begins to curve upward as \(x\) increases.
  • At \(x = 1\), \(\ln 1 = 0\).
  • As \(x\) nears zero, the values of \(y\) become increasingly negative.
The behavior of \(\ln x\) shows its growth pattern that starts slow near \(x = 1\) and gets steeper as \(x\) becomes larger. Understanding this growth is crucial for interpreting graphs that involve the natural logarithm function.
Graphing Utilities
Graphing utilities are essential tools for visualizing mathematical functions, such as \(y = \ln x\). These tools allow us to efficiently view and analyze complex graphs without manually sketching them. When you use a graphing utility, you can:
  • Enter the function you want to graph, like \(y = \ln x\).
  • Adjust the viewing window to see different parts of the graph.
  • Visualize the behavior of functions as input values change.
For the natural logarithm function, the graphing utility will display the curve starting at \(x = 1\) with \(y = 0\), showing the asymptotic behavior as \(x\) approaches zero and continuing to rise as \(x\) increases. These visual representations aid in understanding the function's characteristics and solving related inequalities.
Shading Graph Regions
Shading graph regions is a technique used to illustrate solutions of inequalities, such as \(y < \ln x\). With inequalities, the goal is to represent areas on a graph where the inequality holds true. For \(y < \ln x\):
  • You'll focus on all points where the graph of \(y = \ln x\) is above the shaded region.
  • Shade the area below the logarithmic curve.
  • The line itself should be dashed to indicate that it is not included in the region.
By shading the region correctly, you visually demonstrate every point where \(y\) is less than \(\ln x\). This method of graphing provides a clear, visual understanding of which \(x\) and \(y\) values satisfy the inequality, making it easier for observers to verify solutions.

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Most popular questions from this chapter

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