91Ó°ÊÓ

The natural logarithm, represented as \( \ln(x) \), is a logarithm to the base \( e \), where \( e \) is an irrational number approximately equal to \( 2.71828 \). It is commonly used in mathematics due to its unique properties related to exponential growth and decay.
The function \( y = \ln(x) \) has its own distinct characteristics:

• It is only defined for positive values of \( x \). This means \( x \) must be greater than zero.
• The graph of \( y = \ln(x) \) passes through the point (1,0) because \( \ln(1) = 0 \).
• As \( x \) increases, \( y \) also increases without bound, although it does so at a decreasing rate.
• As \( x \) approaches 0 from the positive side, \( y \) approaches negative infinity.

Understanding these properties is crucial when tackling problems involving natural logarithms, such as graphing inequalities.

Graphing inequalities, like \( y \leq \ln(x) \), involves several steps similar to graphing equations, but with additional considerations for the inequality.
The process begins with graphing the related equation, in this case \( y = \ln(x) \). This gives us a visual reference for the inequality.
It's important to note the direction of the inequality:

• "Less than or equal to" (\( \leq \)) means we will consider all points below and on the line.
• This is represented by a solid line in the graph, indicating that points on the line are included in the solution.

Understanding how to graph inequalities accurately is essential for correctly identifying solutions.

Shading regions on a graph is a visual technique used to represent the area where an inequality holds true.
For the inequality \( y \leq \ln(x) \), the area to shade is everything below and including the curve \( y = \ln(x) \).
To correctly shade this region:

• Start by lightly shading the area directly below the curve.
• Ensure the shading extends to all parts of the graph where \( x > 0 \) since the natural logarithm is only defined for positive \( x \) values.
• The boundary line, \( y = \ln(x) \), should be kept solid to show that points on the line are included in the solution set.

Shading helps visually represent solutions and is a useful strategy in solving graphing problems.

Graphing a nonlinear inequality such as \( y \leq \ln(x) \) involves distinct steps:1. **Understand the Inequality**: Identify the function and the direction of the inequality.2. **Sketch the Curve**: Begin by graphing the equation \( y = \ln(x) \). This curve is essential for understanding where the inequality applies.3. **Apply the Inequality**: Use the inequality \( y \leq \ln(x) \) to determine which regions will be shaded. This often means shading below the curve for a "less than or equal" inequality.4. **Shade the Region**: Carefully shade the valid region, ensuring the boundary is included if the inequality is "less than or equal to."5. **Check Conditions**: Verify that the shaded area only exists for valid \( x \) values, such as \( x > 0 \) for \( \ln(x) \).
By following these steps, anyone can accurately graph nonlinear inequalities.