Problem 108 - Precalculus

Question

Determine whether each statement is true or false. The horizontal axis is the horizontal asymptote of the graph of \(y=\ln x\).

Step-by-Step Solution

Answer

The statement is false; there is no horizontal asymptote for \( y = \ln x \).

Step 1: Understand the Function

We begin by considering the logarithmic function given, which is \( y = \ln x \). This is the natural logarithm function.

Step 2: Identify Properties of the Natural Log Function

For the function \( y = \ln x \), the domain is \( x > 0 \) and the range is all real numbers. As \( x \) approaches zero from the right, \( \ln x \) approaches negative infinity, and as \( x \) increases towards infinity, \( \ln x \) increases towards infinity.

Step 3: Analyze Asymptotic Behavior

An asymptote is a line that a graph approaches but never touches. For horizontal asymptotes, we consider the behavior of \( f(x) \) as \( x \to \infty \) or \( x \to -\infty \). Since \( y = \ln x \) continues to grow indefinitely as \( x \to \infty \), there is no horizontal asymptote.

Step 4: Conclusion on Horizontal Asymptote

Given the analysis, the function \( y = \ln x \) has no horizontal asymptote because it does not settle to a horizontal line at any point along the horizontal axis. The statement that 'The horizontal axis is the horizontal asymptote of the graph of \( y=\ln x \)' is false.

Key Concepts

Horizontal Asymptotes

In mathematics, horizontal asymptotes are horizontal lines that a function approaches as the input, or "x" value, becomes very large or very small. Specifically, if as \( x \to \infty \), \( f(x) \) approaches a number \( L \), then \( y=L \) is the horizontal asymptote. Similarly, if \( x \to -\infty \) and \( f(x) \to L \), \( y=L \) is a horizontal asymptote.

For the function \( y = \ln x \), the behavior is quite distinct. As \( x \to \infty \), the natural logarithm \( \ln x \) grows without bound, moving off toward infinity rather than settling towards a particular horizontal line. Likewise, since the logarithm function is undefined for \( x \leq 0 \) and approaches \( -\infty \) as \( x \to 0^+ \), it doesn't get confined near any horizontal line in negatives either.

Thus, \( y = \ln x \) does not have a horizontal asymptote. Instead, it has a different behavior as it continues to increase indefinitely as \( x \) increases, and drops singularly as \( x \) nears zero from the positive side.

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is a fundamental mathematical function frequently used in calculus and various scientific fields. It is the logarithm to the base \( e \), where \( e \) is an irrational number approximately equal to 2.71828. This constant "e" forms the base of natural logarithms and is a unique number appearing in various contexts in mathematics.

The natural logarithm has several key properties:

• Domain: The natural logarithm is defined only for positive values of \(x\). This means that you cannot take the natural log of zero or a negative number.
• Range: The output of \(\ln x\) can be any real number, positive or negative. As \(x\) increases, \(\ln x\) grows larger and larger without bound.
• Behavior: As \(x\) approaches 1, \(\ln x\) approaches 0. It passes through (1,0) on a graph. When \(x\) is close to zero, \(\ln x\) drops steeply down towards negative infinity.

Understanding the properties of the natural logarithm helps in examining its graph and applications in real-world problems.

Graphical Analysis

Graphical analysis involves studying graphs of functions to understand their behavior visually. For the function \( y = \ln x \), the graph presents several distinctive features that become evident when plotted.

When you draw the graph of \( y = \ln x \), you will observe the following:

• Intercept: It crosses the y-axis at the point \((1, 0)\), since \( \ln 1 = 0 \).
• Behavior near the origin: As \( x \to 0^+ \), \( \ln x \to -\infty \), showing a steep drop.
  
• Growth pattern: As \( x \to \infty \), \( \ln x \to \infty \), the graph keeps rising without stabilizing, indicating no horizontal asymptotes.

By recognizing these characteristics, one can better grasp the dynamics of the natural logarithm. This insight clarifies why the thought of a horizontal axis as an asymptote in this case is false. The graph illustrates infinite and unbounded growth, unlike cases involving horizontal asymptotic behavior.