Problem 19 - Precalculus with Limits

Question

In Exercises 19-28, use a graphing utility to graph the inequality. $$y<\ln x$$

Step-by-Step Solution

Answer

The graph of the inequality \(y < \ln x\) will look like the graph of the function \(y = \ln x\) but also include the area below the curve. Solution area will be shaded indicating all values of \(y\) that are less than \(\ln x\).

Step 1: Understanding the Equation

Firstly, graph the equation \(y = \ln x\) to get a curve with its starting point at \(x = 1\) and \(y = 0\). The curve will continue to rise as \(x\) increases.

Step 2: Graphing the Inequality

Now, considering the inequality, it's clear from \(y < \ln x\) that the solution set consists of values of \(y\) that are less than the values determined by \(y = \ln x\). In other words, the solution set lies below the curve produced by the logarithmic function.

Step 3: Distinguishing the Solution Area

To clearly indicate the solution region on the graph, it's common to shade the region below the curve, which represents all the points where \(y\) is less than \(\ln x\).

Key Concepts

Graphing Utilities

Graphing utilities are digital tools that help you visualize mathematical equations and inequalities. These tools can come in different forms, such as graphing calculators or software applications. They plot graphs based on the equations you input, helping you understand complex relationships between variables.

Using a graphing utility to plot an inequality like \(y < \ln x\) involves a few steps:

• First, you input the equation of the boundary line, which in this case is \(y = \ln x\).
• The tool will generate a graph of this line, showing you the boundary of your inequality.
• Next, you tell the utility to show all the possible \(y\) values that satisfy the inequality, creating a shaded region below the line.

Employing these utilities not only saves time but also reduces the potential for human error in calculations, making them invaluable for students learning to graph inequalities.

Logarithmic Functions

Logarithmic functions, such as \(y = \ln x\), play a crucial role in various mathematical applications. This type of function represents the inverse of exponential functions. The natural logarithm denoted by \(\ln\) uses the constant \(e\) (approximately 2.718) as its base.

Here are some key properties of logarithmic functions:

• The curve \(y = \ln x\) starts from the point \((1, 0)\), meaning the logarithm of 1 is 0 because \(e^0 = 1\).
• As \(x\) increases, \(y\) also increases but at a decreasing rate, creating a smooth upward curve.
• Logarithmic functions are only defined for positive values of \(x\), which helps in defining the domain of the function.

By understanding these properties, you can better predict the behavior of the logarithmic curve, which is essential when identifying solution regions for inequalities involving \(\ln x\).

Solution Regions

When graphing inequalities like \(y < \ln x\), it's important to find the correct solution region on the graph. The solution region represents all the points satisfying the inequality. For the inequality \(y < \ln x\), this region is below the curve of the logarithmic function.

Here's how you identify and represent the solution region:

• First, plot the curve \(y = \ln x\) using a graphing utility. This curve acts as a boundary between the solution and non-solution regions.
• Next, because the inequality symbol is "<", identify all the points below the curve where the inequality is true.
• To visually distinguish this region, shade the area below the curve on the graph. This shaded area is where every point satisfies the condition \(y < \ln x\).

This process helps clearly convey where solutions to the inequality lie, which is crucial for solving and interpreting results in various mathematical problems.