Problem 3 - Intermediate Algebra

Question

Fill in the blanks. The graph of \(f(x)=\log _{2} x\) approaches, but never touches, the negative portion of the \(y\) -axis. Thus the \(y\) - axis is an _____ of the graph.

Step-by-Step Solution

Answer

The \(y\)-axis is a vertical asymptote of the graph.

Step 1: Identify the Characteristics of the Function

The function given is \(f(x) = \log_{2} x\). This function is a logarithmic function with a base of 2. One key characteristic of logarithmic functions is that they have a vertical asymptote at \(x = 0\).

Step 2: Understand the Behavior of the Graph

For \(f(x) = \log_{2} x\), the graph approaches the line \(x = 0\) but never touches or crosses it. As \(x\) approaches 0 from the right, the value of \(f(x)\) decreases without bound.

Step 3: Define an Asymptote

An asymptote is a line that the graph of a function approaches but never actually reaches. In this context, the graph of \(f(x) = \log_{2} x\) gets closer and closer to \(x = 0\) as \(x\) becomes smaller from the right side.

Step 4: Fill in the Blank

The definition of an asymptote fits the description provided in the problem, which states that the graph approaches, but never touches, the negative portion of the \(y\)-axis. Therefore, \(x = 0\), which is the \(y\)-axis, is the vertical asymptote for this graph.

Key Concepts

Vertical Asymptote

In the study of logarithmic functions, a vertical asymptote is a line that the graph gets infinitely close to but never quite reaches. For the function \( f(x) = \log_{2} x \), this vertical asymptote occurs at \( x = 0 \). When you think about asymptotes, imagine a boundary that a function cannot cross.

For any logarithmic function \( f(x) = \log_b x \), the vertical asymptote is typically the \( y \)-axis or \( x = 0 \). However, the graph only approaches this line from one side, the positive side, since the logarithm of a non-positive number is undefined.

This characteristic mainly occurs because, as \( x \) gets closer to zero from the right (positive values of \( x \)), the function's value becomes infinitely negative. Thus, this explains why these graphs always have a vertical asymptote when dealing with natural or real-world data.

Graph Behavior

Graph behavior in logarithmic functions can be intriguing. A logarithmic graph, such as \( f(x) = \log_{2} x \), never actually intersects or touches the vertical asymptote. Instead, it steadily approaches it. As a reminder, asymptotes are theoretical lines that a graph nears but does not cross.

Here's what happens specifically:

• The graph moves closer to the \( y \)-axis but constantly stays on the right side of it.
• As \( x \) gets greater than one, the graph keeps ascending but at a decreasing rate.
• When \( 0 < x < 1 \), the graph dives downward quickly, approaching negative infinity without reaching zero.

These features depict a classic logarithmic curve, which becomes evident on most common graphing devices or software you might use. Understanding this behavior helps in recognizing and predicting the pattern and dynamics of logarithmic functions.

Base of Logarithm

The base of a logarithmic function significantly affects the function's appearance and properties. Logarithmic functions have different forms, such as \( \log_b x \), where \( b \) is the base. In our case, \( f(x) = \log_{2} x \), the base is 2.

Here's how the base impacts the function:

• The base determines the rate of increase. A larger base means a slower increase in function value for increases in \( x \).
• For example, with \( \log_{2} x \), for \( x = 2 \), the function value is 1 since \( 2^1 = 2 \).
• If the base were changed to a number greater than 2, the graph would rise less steeply.

To explore these variations, you can experiment by changing the base in graphing software. Observing different bases will deepen your understanding of how these functions behave dynamically. Always remember that the base must be greater than 0 and cannot equal 1 for \( \log_b x \) to be valid and meaningful.